Here is a somewhat speculative idea that recently occurred to me. I don’t think I will be able to deliberate on all the comparative implications just now, but it wouldn’t surprize me too much if something similar had already been proposed.
A relatively well-known suffix element usually reconstructed for Proto-Finno-Ugric (implicitly Proto-Uralic, in wake of Finno-Ugric turning out to be probably not a genetic grouping) is the comparative suffix *-mpa, reflected quite well in the best-known Uralic languages: Finnish -mpi : -mpa-, Estonian -m, Hungarian -(a)bb. [1] The Samic languages also have clear cognates, e.g. the Northern Sami bisyllabic adjectives’ comparative ending -t : -bu- (from slightly earlier *-b : -bu-).
No trace of such a comparative suffix though is known elsewhere in Uralic. This smells slightly suspicious. Hungarian is, overall, quite innovative, and usually whatever clear old Uralic features have been retained there, can also be traced in at least some of its Russian relatives; especially Ob-Ugric and Permic. There’s a proposed Samoyedic cognate, but as far as I recall seeing, found only in Nenets — and only used in an approximative sense.
It’s also the case that there is a quite well-established PU participle ending *-pa. These two suffixes share the privilege of being just about the only places in comparative Uralic inflectional morphology where *p occurs; and both of them have very roughly adjectival semantics. Might it be possible to thus segment *-mpa as *-m-pa? We’d like to know if this can be made to make semantic sense; and if we can find a reasonable candidate for what the nasal element comes from.
The former question can be roughly reformulated as: “if a thing is greater (than another one), what is it doing?” To me it seems the answer would be “exceeding, being greater”. A PU “comparative” form such as *wod₂ə-mpa (> Fi. uude-mpi, Hu. új-abb ‘newer’) could then be instead analyzable as *wod₂əm-pa, meaning ‘that which *wod₂əN-s’; and which could have independently developed into an IE-style nominal comparative in Hungarian and Finno-Samic. Originally it’d have been instead the verb stem *wod₂əN- that captured the “comparativeness”, meaning something like ‘to be newer’. The approximative sense in Nenets seems well-derivable from this as well — we can easily imagine the base meaning as just ‘to be new’, and derive from this on other hand an amplification ‘to be newer’, on the other hand a mitigation ‘to be newish’.
Above I’ve written the nasal of my internally reconstructed verb stem as just -N-. While Proto-Uralic allowed heterorganic nasal+stop consonant clusters with a coronal stop, [2] there seem to be no examples of this with a peripheral stop. Only *mp and *ŋk can be reconstructed stem-medially, while there are no **np, **nk, **ŋp, **mk. So I suppose all nasal consonants are fair game here. (And, of course, in Finnic and Hungarian all such heterorganic clusters assimilate anyway.)
— Now consider Finnish verbs derived with the suffix -ne-: e.g. iso ‘big’ → isone- ‘to become bigger’; mätä ‘rotten’ → mätäne- ‘to rot’; pimeä ‘dark’ → pimene- ‘to become dark(er)’. The suffix is used almost exclusively on adjectives, and typically forms verbs meaning indeed increase in quality. This seems to provide a great candidate for a Proto-Uralic derivative class, on which the nominal-type comparatives of “Western European Uralic” could have been based. Altogether, originally a word like Fi. isompi ‘bigger’ would have been a consonant-stem participle, equal to modern Fi. isoneva ‘that which increases, becomes bigger’ (pseudo-PU *ićäwmpä ~ *ićäwnəpä).
A chief remaining problem would be whether we really can reconstruct this verbal suffix all the way to PU though. SKRK reports similar usage across Finnic, as well as possible cognates in Ob-Ugric and Samoyedic; these however indicating original *-m-! Another hypothesis mentioned would be comparison to a momentane suffix -n in Hungarian, found in some fossilized forms such as villan ‘to flash’ (seemingly related in some way to világ ‘world; (archaic) light’ < PU *wëlkə). This sounds a bit better with respect to my reconstruction, but I’d like having some more supporting evidence. And of course, I’d also have to check how well the development of comparison constructions elsewhere in Uralic can be lined up with this scenario.
[1] For clarity, I’m ignoring vowel harmony in this post.
[2] Perhaps the clearest case is *tumtə- ‘to know’, whence e.g. Fi. tuntea, NS dovdat, Hu. túd, Tundra Nenets tumtă-.
Freelance reconstruction 
Postscript: I should perhaps also note that the idea that *-mpa was not originally comparative in meaning is not especially new. E.g. Janhunen in “On the structure of Proto-Uralic” (1982, FUF 44) suggests a vague meaning of “local contrast”.
“it wouldn’t surprize me too much if something similar had already been proposed.”
In fact, the hypothesis that the Finnic comparative suffix *-mpa is originally a participle of verbs with the suffix *-ne- was proposed by D.V. Bubrikh in his posthumously published book “Историческая морфология финского языка” (“Historical morphology of the Finnish language”, first published in 1956). Bubrikh also noted that comparatives derived from a-stems replace the /a/ with /e/ in exacly the same way as verbs with *-ne- suffix. Moreover, the fact that this replacement occurs only in comparatives derived from disyllabic stems, was connected by him with the fact that *-ne-verbs are derived only from disyllabic nouns.
Unfortunately, since Bubrikh wrote when it was risky to engage in comparative linguistics (Marrism being the official doctrine in the Soviet Union), he almost completely restricted his work to internal reconstruction. So we do not know his thoughts on extra-Finnic comparanda for *-ne-verbs.
The reference for Bubrikh’s work is:
Д.В. Бубрих “Историческая морфология финского языка в связи с синтаксисом” // Д.В. Бубрих. Прибалтийско-финское языкознание: Избранные труды. СПб.: Филологический факультет СПбГУ, 2005.
The hypothesis in question is discussed on pp. 202-203.