I have a feeling this is better suited to r/asklinguistics, but I’m answering anyway because it’s just so wonderfully specific.

The short answer is: no, according to everything we know, Russian retained its cases even amongst peasants. Dialectal differences were mostly in terms of pronunciation and vocabulary.

The long answer:

It’s true that the Russian language, as is often the case, was standardised around the variety spoken by the intellectual élite, specifically the one spoken in Moscow and large cities in Central Russia. Likewise, it’s true that there wasn’t a real attempt at enforcing this standard until Soviet education programmes. Russian dialects in general had always been largely ignored, with most scholars focusing on folklore and traditions rather than vocabularies and grammars. When a widespread education system was established those dialects were dismissed, made fun of or straight up eradicated in favour of the standard. Up until the ‘90s, radio and then TV hosts were taught to speak in the Moscow dialect. This is one of the reasons why Russian is so homogeneous despite the country itself spanning from the Baltic to the Sea of Japan.

That doesn’t mean the Russian dialects are extinct. There’s still a certain degree of variations, evident mostly on the European side. Besides little changes in vocabulary, those variations mostly have to do with pronunciation. We have linguistic phenomena such as akanje, where unstressed Os are pronounced as [a] rather than [ɐ] or [ə], and its opposite okanje, where unstressed Os are rendered as [o] with little to no change. Depending on where you go you might find t’ being more or less palatalised, or standard [g] being pronounced as [ɣ]. Speakers in Saint Petersburg retain [t͡ʃ] in most consonantic clusters, contrary to Moscovites who, much to the formers’ gripe, pronounce it as [ʃ].

As far as we’re able to tell, this was the case in post-revolutionary Russia as well. Peasants largely spoke in their own dialect, which at times could vary greatly from the standard, but, again, mostly in terms of pronunciation. If you read any Russian novel from the 19th century you might find peasants shortening patronyms, so that for example Antonovič becomes Antonyč. Indeed shortening words or dropping final vowels or consonants is one of the most commonly found features. This was of course seen as a mistake, especially when combined with switching noun endings (a instead of ы in the genitive case, for example), and it presented all sorts of problem when learning how to write, especially in light of the ortographical reform of 1918. There is however no indication that Russian peasants spoke without cases. If they did, that would have been basically a different language, something that would have probably sparked the interest of scholars at the time.

A final note on Vulgar Latin: despite its name, Vulgar Latin wasn’t spoken by the lower classes in Ancient Rome. The term refers to the varieties of Latin spoken in Europe after the collapse of the Roman Empire which would eventually develop into the Romance languages (French, Italian, Spanish, and so on). Those varieties did in fact lose most cases, but it was a matter of language(s) evolving through time rather than social differences.

It is a huge myth that "Vulgar Latin" "had no noun cases." The spoken varieties of Latin certainly did have noun inflections. Medieval French and Occitan preserved a dual case system into the Middle Ages in writing consisting of the nominative and an oblique, and Romanian has a nominative-accusative and genitive-dative combined endings today. In fact when Classical Latin was standardized it had the same case endings spoken Latin did because Classical Latin WAS the "Vulgar" Latin of upper-class Romans. It was over time the case system slowly collapsed. Between the standardization of Classical Latin in the 1st century BC and 2nd Century AD the vocative disappeared, and the accusative and ablative merged together into one ending either because of loss of final -m, making some accusative and ablative endings the same, or one or the other became the preferred to use as the object of the sentence or with prepositions. This left us with a 4-case system that probably by the 5th century AD was reduced to a 3-case system in vernacular Latin. This 3-case system was the nominative, a shared/merged together genitive-dative, and the accusative-ablative. By the time of the early 7th century AD Italo-western romance most likely had the dual nominative and oblique like Old French and somewhere between then and a bit before Romance texts are attested Italy and Spain loses the nominative.