I don't know much about the Beatles, but I know they had a few movies.
So, how about this. In an attempt to boost ratings, NBC convinces Gene Roddenberry to replace one of thsoe drab Seson 3 episodes with a Beatles one, becasue somehow the Beatles have learned about Star Trek and Ringo has become a big fan. Becasue Ringo is the only one who really cares, he is given the majority of the lines for the 23rd century spacefaring rockers.(1)
Something about Starr's character resonates with Trekkies, athe show gets a rebirth in the '70s through reruns, Starr becomes moderately enthused about the conventions. Paul is with Wings by this time, Lennon is Lennon, and I don't know what George is up to, but needing something to do after the breakup, Ringo gets involved and even does a voiceover in a couple episodes of The Animated Series.
Soon, Ringo Starr finds himself somewhat typecast, but he is doing better than he ever could have musically, and after John Lennon's death, Paul - recognizing the Beatles can't "get back together" anymore, readily admits that, "Ringo has made more of himself than one ever would hve expected, and I certainly wish him the best."
Ringo Starr ends up as a recurring guest star playing a few different characters on ST:TNG(2) - including in an episode about 4 people awakened from cryogenic sleep after 300 years, one who happens to be a musician and who has some interesting moments for the fandom as he discusses with Data the music of the 24th century(3) - and also ends up in the later Doctor Who series. In fact, there is continued speculation on the internet about him being key to a crossover movie, though it never takes place except in fan fiction.
okay, not as fun as
my Anissa Jones one in Shuffling the Treks, but they say "write what you know," right?

Others can decide how much Ringo Starr could do. And, it is actually a plausible POD since OTL, if I recall, Trek wasn't in its first season in Britain till at least its 3rd season in the US. So, if he sees it earlier, who knows?
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(1) In a series that had space hippies, why couldn't it happen?
(2) You were hoping for him taking Patrick Stewart's place? This isn't the ASB forum
(3) Originally the episode had 3, IIRC - interesting episode, too, I thought.