Well, I guess it depends on what you mean by "impact." On the one hand, "socialist realism" probably didn't have a lot of meaning to the average person in Russia. However, it did become the dominant form in the arts, such as it was, and people would have been exposed to it. It also became a standard for censorship which, again depending on how you are framing the question, prevented the average person from being exposed to other kinds of art forms.

I assume because you are asking the question you have a decent understanding of what socialist realism was, so I won't go into that too much but I should say a few things that are relevant to your question. As you might know it was really a response to the revolutionary/experimental art forms that popped up after the revolution to try to answer the question "What is socialist art?" All sorts of forms were tried. Setting aside personal preference, the critique that many of these were unintelligible to to the average person was true. Denise Youngblood's Movies for the Masses shows that despite the (genuinely) high minded ideals of many artists, they "artsy" films were often poorly attended.

It's also worth noting that "socialist realism" was never really particularly well defined. Sergei Eisenstein's notions of montage were somewhat radical/avant-garde in the 1920s. These sorts of films (like Strike or October) were the kind of thing that were not made when socialist realism was enforced. Nonetheless, if you actually read Eisenstein's thoughts on the subject, he embraced socialist realism and didn't necessarily see it as totally in contradiction to his ideas of montage (there are English translations of many of his writings available in two books called Film Form and Film Sense). Now, thinking critically about that, we might say that he was compelled to say that because of the political ramifications of rejecting socialist realism. Nonetheless even in his films from the 1930s like Alexander Nevsky you can see some of his earlier ideas present even thought they are not as ubiquitous. This was only really possible because socialist realism didn't have a very clear definition.

All of that is to say, that when a film like Alexander Nevsky was seen by millions (an incredibly popular film), it certainly made an impression on people. But your average viewer was not considering the use of montage and how that might or might not contradict "socialist realism." Nonetheless, socialists realism's goal of promoting communism was certainly achieved in the film, which might suggest that the form at least in some cases did have it's desired impact on the population.