There are, I think, several things off about the current understanding about the treatment of the consonant clusters *wr and *wj in Proto-Finnic.
There are no generally accepted instances of *-wr- in Proto-Uralic (though see below for one proposal), and examples with *-wj- are rare enough that so far none of them happens to have Finnic reflexes (probably the most reliable is *jäwjə ‘beard lichen’, with reflexes in just three branches: Samic + Khanty + Samoyedic). Within the Finnic comparative data, no direct evidence for these clusters appears either.
Cases involving these clusters in Proto-Finnic are therefore solely Indo-European loanwords. In these, two different lines of treatment have been generally accepted.
The first is metathesis to *-jw-, *-rw- > *-iv-, *-rv-. These latter clusters clearly occur in material inherited from pre-Finnic (e.g. PF *kaiva- ‘to dig’ ~ Samoyedic *kajwå ‘spade’; PF *sarvi ‘horn’ ~ Samic *ćoarvē ‘id.’). One classic example of the metathesis of *-wr- has been known for centuries: the word for ‘lake’, PF *järvi ~ PS *jāvrē.  Among Baltic loanwords, about three other examples can be found: *karva ‘hair’, *tarvas ‘bull’ and *torvi ‘horn (instrument)’ (~ e.g. Lithuanian gauras, tauras, ‘id.’; Latvian taure ‘id.’). ‘Lake’ has been proposed to be a loan as well, except from an earlier stage of Balto-Slavic, to account for reflexes also in Mordvinic and Mari.
Cases of metathesis of *-wj- are a newer discovery. Germanic *-wj- being continued as Finnic *-iv- was established some decades ago by Koivulehto,  with examples such as *laiva ‘ship’ ← Gmc. *flawją ‘id.’ (> e.g. Old Norse fley); *raivat- ‘to clear out, esp. woodland’ ← Gmc. *strawjan- ‘to strew’; *raivo ‘skull’ ← Gmc. *trawją ‘vessel’ (ONo treyja). Examples in loanwords from other sources seem to be rare so far, but one is the Estonian rivername Koiva, located in northern Latvia; and whose Latvian name is instead Gauja.
The other development is fortition to *-pj-, *-pr-. Both of these are clusters introduced in loanwords in the first place, and examples of this development are generally later loanwords from Germanic. Examples are not too numerous, but they include *hipjä ‘skin’ ← Gmc. *hiwją ‘appearence’ (ONo hý); — *hapras ‘brittle, weak’ < *šapras ← Gmc. *sawraz ‘filth, dirt’ (ONo saurr); *sapra ‘a type of haystack’ ← Gmc. *sauraz ‘pole’ (ONo saurr); *äpräs ‘bank, steep shore’ ← Gmc. *awriz or *awraz ‘sandbank’ (ONo eyrr, aurr).
I do not aim to question any of these etymological correspondences. However, I find the idea that both developments would have arisen specifically as sound substitutions to avoid the “phonotactically forbidden” clusters *-wj-, *-wr- implausible.
There is one principal problem. While the Proto-Finnic period involved a hefty reduction in the total consonant inventory of the language (loss of palatalized *ć *ś *ń, the “spirants” *d₁ *d₂ *x, the postalveolar affricate *č and the velar nasal *ŋ), it on the other hand brought a clear increase in phonotactic complexity. Some new types of consonant clusters that appear to have been introduced include:
- stop/affricate + liquid, e.g. *sëpra ‘company’, *atra ‘plough’, *ocra ‘barley’, *nakris ‘turnip’; *täplä ‘spot’, *kakla ‘neck’
(no *tl though)
- stop + nasal, e.g. *litna ‘town’, *sakna ‘sauna’;
- stop + *j, e.g. *kapja ‘hoof’, *patja ‘mattress, pillow’, *acja ‘thing’, *vakja ‘wedge’;
(also, in native vocabulary, *-tv- < *-d₂w-, in e.g. *patvi ‘tinder’;)
- fricative + nasal, e.g. *käsnä ‘callus’, *lehmä ‘cow’, *ahnas ‘ferocious’;
- fricative + semivowel, e.g. *rasva ‘fat’, *ohja ‘guide’, *rahvas ‘people’;
- nasal + geminate stop, e.g. *temppu ‘trick’, *kontti ‘leg; backpack’, *lonkka ‘hip’;
- liquid + geminate stop/affricate, e.g. *harppat- ‘to take a long stride’, *kartta- ‘to avoid’, *tarkka ‘acute, accurate’; *hëlppo ‘easy’, *hëltta ~ *helttä ‘cockscomb’, *malcca ‘Atriplex sp.’, *palkka ‘salary’;
(found through inflection and derivation also in native vocabulary, e.g. *jält-tä, partitive sg. of *jälci ‘cambium’)
- liquid + affricate, in at least *porcas ‘pig’;
(found also in native vocabulary through *-Rtə >> *-Rci)
- liquid + fricative, e.g. *varsa ‘foal’, *vërho ‘drape’, *kulha ‘bowl’;
- *n + fricative, e.g. *pënsas ‘bush’, *vanha ‘old’;
- geminate nasal, e.g. *konna ‘toad’;
- geminate liquid, e.g. *villa ‘wool’.
(from earlier *ln)
This was not a momentaneous revolution in phonotactics, of course. For a few of these, examples of rather uncertain Uralic derivation have been suggested (e.g. ‘turnip’ has been compared with Mansi *nëër, Khanty *naaɣər ‘pine nut’); others have been introduced in relatively early loanwords and have thus “non-native cognates” elsewhere in Uralic (e.g. Mordvinic *purćəs ‘pig’); others may not have yet been introduced in Proto-Finnic proper, but rather in some of the early Finnic dialects (such as *käsnä, found only in Northern Finnic). None of this rocks the overall picture, though: if loanwords were able to feed in new types of clusters, they were taken up as-is, just about as far as possible.
(The same process has also kept going later on. Even in varieties such as standard Finnish, where there has been no post-Proto-Finnic syncopë to generate new native clusters, the ongoing flow of various Indo-European loanwords still has by now introduced loads more of novel consonant clusters, such as /-stm-/ in astma, /-ŋ(k)st-/ in gangsteri, /-kstr-/ in ekstra.)
So why would *-wj- and *-wr- have been specifically and stubbornly avoided for centuries? Especially when this general type of cluster, semivowel + sonorant, was able to occur in native vocabulary all along, as is shown by e.g. Fi. läyli ‘heavy’ < PF *läüli < PU *läwlə, or the above-mentioned Fi. kaivaa ‘to dig’ < PF *kaiva- < PU *kajwa-.
I propose that the main part of the solution is that the the alleged “metathesis upon substitution” did not quite occur. This was instead a regular sound change, one that merely happened to mainly operate on loanwords.
Some indirect support is provided, I think, by how other examples of continuant cluster metatheses are already known in Finnic, too. These include:
- *-jh- > -hj- in North Estonian (lahja ‘thin, lean’ ~ Fi. laiha)
- *-nh-, *-lh-, *-rh- > -hn-, -hl-, -hr- in South Estonian (vahn ‘old’, võhl ‘witch’, kahr ‘bear’ ~ Fi. vanha, velho, karhu; NEs. vana, võlu, karu)
- *-wh- > -hv- in both NEs. and SEs. (kehv ‘poor’ ~ Fi. köyhä)
- *-sn- > -ns- in Western Finnish (runsas ‘plentiful’ ~ Livvi ruznaz)
For the metathesis *-wj- > *-jw- in particular there is also an areal parallel from Ter Sami and Lule Sami (you may recall I have already mentioned the Finnic metatheses currently under discussion in that post, too).
Since these metatheses affect only a part of the Finnic (or Samic) languages, sound change seems to be the only explanation available. It is not clear to me why continuant clusters would be particularly prone to metathesis though, and it’s possible that e.g. the connecting factor in the first three changes could be the metathesis of *h specifically. Regardless, it seems rather arbitrary to instead prefer a sound substitution explanation for *-wj- and *-wr-.
A number of the individual words to have been metathesized specifically point towards a sound change rather than a sound substitution, too.
1) For ‘lake’ there are two possible arguments. The first is chronological: *jäwrä could be regularly reconstructed already for Proto-West-Uralic (or for Proto-Finno-Volgaic, if you were to subscribe to such a stage). Alternately, there may be a phonetics argument available. Research in Uralic substrate vocabulary in Western Russia has led to supposing a “Meryan” reflex *jäkr- as well, reflected in lake names with an element ягр- or яхр-.  Both phonetic typology, and the proposed early Balto-Slavic etymology of this whole ‘lake’ root (either from *yewH-ro- ‘body of water’ ; or from *eǵʰe-ro- ‘lake’? ) suggest that the velar element in these has not developed from *w, but is instead an archaism, pointing to *jäkrä or *jäxrä as the earliest shape of the word in Uralic, with lenition to *-wr- at least in pre-Finnic and pre-Samic.
None of this is still completely watertight though, as another possibility yet is that early BSl. *-wHr- was substituted as *-Kr- in pre-Meryan, but as *-wr- at least in pre-Samic. If so, loaning to Proto-Finnic could have happened independently as well. (Mordvinic and Mari only show simple *r and they can swing any way really.)
2) With ‘horn’, the appearence of *o in Finnic may point to *towrə as an earlier form. This ties in with a larger topic: several Baltic as well as some Germanic and Indo-Iranian loanwords in Finnic seemingly still preserve PIE *o — but in a few of the cases we are actually dealing with PIE *a instead, one of these being this particular word (East Baltic *taure is, obviously enough, a derivative of *tauros ‘bull, aurochs’). I’ve prepared a small survey of this matter some time ago,  and among other results it turns out that cases with *au → *ou seem to be especially frequent. I suppose that this indicates that Proto-Baltic or Proto-Balto-Slavic had already merged short *a and *o at the time, but that the diphthong *au was during the time realized in some applicable dialect with a labialized first component, roughly [ɒu]. The loanwords with *au → *au would then have to be analyzed as later (as is already the case as well in explanations that appeal to the late retention of PIE *o), or as coming from a different Baltic dialect.
3) The above argument applies almost intact also to Koiva, for which we can likewise posit *Koiva < *Kowja ← *G[ɒu]jā. Here original pre-Balto-Slavic *ou can be suspected as well, though.
4) With ‘ship’, assuming metathesis as a sound law seems to provide a small improvement for the historical phonology of Livonian. In native vocabulary and sufficiently old loanwords, the development of *-Viv- in Livonian is initially *-Vuv-, possibly with monophthongization in modern Courland Livonian (well paralleled by known developments such as *-Vll- > -VVl-, or *-Vlj- > *-Vľľ- > -VVľ-):
- *kaiva- ‘to dig’ > *kauva- > kouvõ : kǭva-
- *koivu ‘birch’ > *kõuvo-s > kȭvaz : kõuv-
- *päivä ‘sun, day’ > *päuva > pǟva
- *taivas ‘heaven’ > *tauvas > tǭvaz (originally from Indo-Iranian)
However, for ‘ship’ we instead find *laija > lǭja : laij-. This could be explained by the metathesis *-wj- > *-jw- having never happened in Livonian. Thus, just as *-jw- > *-Viv- assimilates to *-Vuv-, also *-wj- > *-Vuj- assimilates to *-Vij-; and the development *lawja > *laiva only holds for the rest of Finnic. 
5) Finally, my earlier promised possibly inherited example of *-wr-: *korva ‘ear’.
Older research has taken comparison with Samic *koarvē, approx. ‘prop’ (NS also bealljigoarvi ‘earhole’), Permic *kʷor ‘leaf’, Hungarian dial. harap ‘dry grass’ as grounds to reconstruct PU *korwa ‘blade, leaf’. On semantic grounds, the alleged Samic cognates look like loans from Finnic though. The direct development ‘blade’ > ‘prop’ appears improbable; while the development ‘blade’ > ‘ear’ > ‘handle’ > ‘prop’ (the two last stages are verifiable as polysemic meanings of *korva and its derivatives in Finnic) seems to be etymologically blocked, since Samic still retains the original PU root for ‘ear’, *pealjē < *peljä.
A competing proposal comes from Juha Janhunen, who in ’81 has compared Finnic *korwa with Samoyedic *kåw ‘ear’. In his original opinion, the root here would be approx. *kawə, irregularly labialized in (pre-)Finnic, and extended to a derivative *kow-ra > *korva. Semantically this is clearly better.
I do not find ad hoc labialization in Finnic enticing, though. And there’s also another phonological issue: *kåw is the only Proto-Samoyedic root with a shape *CVw in Janhunen’s reconstruction, while a number of more reliable examples instead point to the regular development being *CVwə > *CV.  Etymologically the proposal has its problems as well. Supposing two synonyms for ‘ear’ with complementary distribution (*kawə in Finnic + Samoyedic, *peljä everywhere else in Uralic) might work under a scenario where Finnic and Samoyedic are two early offshoots of Uralic, but seems less likely if they sort into their own respective wider subgroups, West Uralic and East Uralic (as I think is the most probable).
Despite all these issues, this idea might regardless be onto something. I would instead assume that the original root here is *kow-; and that, while it is not retained as such in any Uralic language, a parallel derivative from this, formed already in Proto-Uralic with the common verbalizing suffix *-l(ə)-, is the well-attested verb for ‘to hear’. This has traditionally been reconstructed as the rather Finnocentric *kuule-, but in my opinion thus better: *kow-lə-.  Several reflexes seem to indicate *o; these include Mordvinic *kuľə-, Mari *kola-, Mansi *kʷaal-, and, if it has any input from here, Hungarian hall- (though Old Hungarian hadl- would seem to show that this is instead from PU *kontV-lə- ‘to listen’). Permic *kɨl- and Khanty *kɔɔL- are the only reflexes compatible with short-vocalic *kulə-, and they might simply result e.g. from a raising *ow > *u, similar to the development *ow > *uu I assume for Finnic. 
It also seems likely to me that the Samoyedic words for ‘ear’ are derived from this root in some fashion, even if probably not as direct inheritance. PU *o > Samoyedic *å is after all the regular development in any environment other than *CoCə. To make progress, I’d suggest that the PSmy reconstruction itself requires adjustment. Janhunen’s monosyllabic *kåw seems to be largely based on Nganasan kou, but this could just as well come from a bisyllabic proto-form such as *kåjå, through the regular loss of post-tonic *-j- and raising of *å (an exact parallel is PU *kaja > PSmy *kåjå > Ng. kou ‘sun’). Given the developent *-wj- > *-j- in *jäwjə > *jüjə ‘beard lichen’, I would prefer assuming an agentive derivative *kow-ja > *kåwjå > *kåjå ‘hearer’. — Or perhaps a more heavily contracted *kowlə-ja > *kol-ja > *kåljå > *kåjå, for an exact parallel with attested forms like Fi. kuulija?  This would even have some strange synergy with the derivation of Smy. *timä < *temä ‘tooth’ from *sewə-mä ‘bite, biting’.
Getting back on track, though. If the metatheses *-wj-, *-wr- > *-jw-, *-rw- took place as regular sound changes in Proto-Finnic times, this will naturally lead to a full absense of *-wj- and *-wr-, as the Finnic comparative data indeed suggests. So far, so good.
However, what from there on? Should we not just as well expect these clusters to be recreated right away by the next few batches of loans, instead of fortition to *-pj-, *-pr-?
At this point I would like to direct attention to the fact that the (Western) Finnish reflexes of these words do not show explicit signs of such a fortition. My example words listed above surface as hauras ‘brittle’, saura ‘haystack’, äyräs ‘bank’, dialectal hiviä ‘skin’ (though Standard Finnish has adopted the fortited form hipiä). This is though indeed also the regular Finnish development of *-pj-, *-pr- (cf. *kapja > kavio ‘hoof’, *sëpra > seura ‘company’)… so, as long as we wanted to route these loans through Proto-Finnic, it will still be preferrable to indeed reconstruct e.g. *hapras, *hipjä, in order to regularly account for all reflexes, including also such ones as Northern Karelian hapraš, hipie.
But consider now the possibility that these aren’t loans dating all the way back to Proto-Finnic; and rather loans acquired after its breakup, taken up in the first place in Western Finnish, and mediated from there to the other Finnic varieties. In this case, the appearence of *-pr-, *-pj- could instead be a type of “etymological nativization gone awry”: e.g. the pre-Karelian dialect would by this time still have remained without **-wr-, but it would have had *-pr- as an equivalent of West Finnish *-wr-. This could have motivated adopting the cluster not phonetically, but rather “phonologically”. 
This firstly allows us to get rid of the strange back-and-forth phonological development in Finnish: words like hauras would simply preserve the Germanic original’s diphthong altogether. Secondly, this allows for some variation in the reflexes elsewhere in Finnic: if different Finnic dialects had to individually deal with adopting West Finnish *-wr- somehow, some of them could have opted for different strategies in different words. And we indeed find a *-wr- ~ *-pr- vacillation in e.g. Fi. teuras ‘sacrificial animal’, teurastaa ‘to slaughter’ ~ Krl. teuraštoa ‘to slaughter’ | Es. tõbras ‘head of cattle’ ~ Votic tõbras ‘elk’. This lexeme is likely from Germanic *þeuraz ~ *steuraz ‘bull’, but no single PF form can be set up. Instead of assuming two parallel loans (*tëpras ‘head of cattle’, *tëuras ‘sacrificial animal’?), it will be possible to reckon with just a single early Finnish loan *teuras, further adopted in differing ways into Karelian and Southern Finnic.
There is one non-trivial cost as well, though. ‘Brittle’ happens to be one of the words showing the characteristic pan-Finnic sound change *š > *h. If the word regardless spread across Finnic by diffusion from dialect to dialect, it will be now fairly difficult to assume that this sound change occurred in unitary Proto-Finnic; it will instead have to be an “areal-genetic” post-Proto-Finnic development. 
I am prepared to defend this dating in detail. *š > *h has already been proposed by multiple researchers to date as later than the split between South Estonian and the rest of Finnic. Drawing it out it further yet would not seem outrageous considering what we know of the typical expansion history of this kind of “major”, i.e. phonologically simple but innovative sound changes — while it would seem to allow the phonological fine-tuning of a handful of other known etymologies as well. But that will have to be a topic of its own.
In case my analysis here is correct (and I think it at minimum should prompt some kind of a more detailed defense for why would there ever have existed a “metathetic sound substitution”), there is a moral to be learned as well. The Finnic languages are often taken as phonologically archaic; this is undoubtedly the case with regards to several features of their inherited lexicon, most prominently the bisyllabic root structure. However, loanwords have been a consistent source of new phonotactic complexity. It is then to be expected that there have been several layers of “renormalization” — processes that have pushed these new root shapes back in line, towards the native word structure. And this may have occasionally swept a few native words along as well. Such innovations will probably be impossible to identify as long as we only look at the native component of the vocabulary, however.
 Although often enough people with preconceptions about the archaicity of Finnic have also assumed that the metathesis was on the Samic side instead — despite how this would have to be irregular: Samic quite well allows *-rv-, as in e.g. ‘horn’.
 Essentially singlehandedly in his 1970 article “Suomen laiva-sanasta“.
 See e.g. Pauli Rahkonen (2011), “Finno-Ugrian hydronyms of the River Volkhov and Luga catchment areas“.
 Most IE cognates seem to point to meanings like ‘river’ or ‘flowing’, but the derivatives in modern Baltic such as Lithuanian jūra ‘sea’, jaura ‘bog’ may have gained this more stationary meaning early on. I wonder if this semantic shift might have originally taken place near the wide and slow-flowing middle parts of the Volga.
 Could it be possible for this root, apparently well-attested only in Balto-Slavic, to be a backloan from Uralic…? It would have to be at least old enough to be pre-Satemization, though, and the “epenthetic” thematic vowel seems hard to explain in this fashion as well.
 You can find a working version over here; written in Finnish though. Maybe I will post an English summary here at some point.
 Dating the assimilation *-jw- > *-ww- very early in pre-Livonian would also work. In this case, newer loanwords could be still subject to the metathesis *-wj- > *-jw-, they would just be later on assimilated in the opposite direction: *-Viv- > *-Vij-. This might be indeed preferrable in light of two other data points. The first is *vaiva ‘bother, trouble, ailment’, which yields Liv. vǭja; it is however a Germanic loanword, whose original seems to require reconstruction with *-jw- (given e.g. Old High German wēwa). The other is the known Livonian developments *-Vlv- > *-ll-, *-rv- > *-rr- (e.g. *sarvi > *sarro > sǭra ‘horn’) taken together, which would surely predict that at this same time *-jv- > *-jj- as well (and not > *-vv-).
 E.g. *śowə > *so ‘mouth’; *sewə- ‘to eat’ > *te-mä > *timä ‘tooth’.
 This would also then disprove the often presented Indo-Uralic comparison with the PIE root for ‘to hear’, *ḱlew-. Instead I believe that better IE comparanda might be √h₂ew- ‘to perceive’ (from which *h₂ōws ‘ear’ is derived); or perhaps *(s)kewh₁- ‘to sense’. (Are these a doublet of some sort?)
 For Khanty, another possibility is that this is from earlier *kʷaal-, as in Mansi; this could have come about as a distant assimilation *kVwC- > *kʷVC-. While speculative, this idea is not quite entirely ad hoc: a possible parallel is *käwd₁ə ‘rope’ > Mansi *kʷääləɣ.
 It would be remotely within possibility to also suggest starting from *korwa or *kowra, as required by Finnic, combined with an ad hoc loss of *r in this cluster. However, I suspect that Finnic *harva ‘sparse, rare’ may be cognate with Samoyedic *tïrå ‘dry’ (< PU *šërwa; cognates in various other branches for both “sides” of this comparison are known as well), which would allow establishing a rather more natural development: PU *rw > Smy. *r.
 This gets perhaps even more phonetically plausible, if we assumed the “cluster series shift” to not have happened immediately from *-wr- to *-pr-, but rather from something like more innovative Western Finnish *-wr- to slightly more conservative Western Finnish *-βr-. This latter cluster would then have had no other option than to be uptaken as *-pr- in pre-Karelian / Ingrian / Estonian / etc. — On the other hand, this would require such a fine-grained Finnish dialect distinction to have indeed existed at the time, which may prove problematic.
 One other technically possible but again contrived explanation would be to assume that the word was initially lost from all Finnic varieties except Western Finnish, and that it later staged a return from there.