When it comes to measuring impact, especially of one piece of work on a large, diffuse audience, there's a lot of methodological difficulty that comes with that. I'll give you my take on the question below (based on teaching Russian history at a SLAC in the US), but let me first approach this question as a historiographical one.
"Rasputin" by Boney-M was released in 1978. By looking at the years that works were published (including many works aimed for a mass audience), we can get a sense of whether there was enough of a boom in interest for authors to try to capitalize on the popularity of the song. Obviously, this wouldn't be instantaneous – it takes time to research, write, and publish a book, and even popular history books would take years – but we might expect to see a jump in the very late 1970s and early 1980s. Here are the post-WWII English-language historiographies, organized by date of publication (as pulled from the UIUC library collection, which has perhaps the most extensive collection of English-language Russianist works):
Based on this list, we can see that there is a slight bump after the song (with two works published in 1982 and one in 1985), but there's certainly no massive increase. After this slight increase, we see a return to the usual 2-3 books published per decade about this topic.
So did the song have any impact beyond this slight jump in historiographical output? Absolutely. Although it's hard to measure in ways that fit the rigorous requirements of this sub, the song has brought the mythos of Rasputin to a huge audience for over four decades. I have at least one or two undergrad students every semester who mention this song as a reason they wanted to study Russian history. On a job interview, the provost of a university started off the interview with the question "What do you think about Rasputin?" because I study early 20th c. Russian history. It is one of many works of English-language popular culture – going back to the film "Rasputin and the Empress" from 1932 – that have kept this mythos alive for over a century.