Assuming the (permuted) Fab Four still take off and get movie contracts, the plot of
Yellow Submarine is quite different--OTL Ringo is the hero of that movie.
Come to think of it he's a big deal in all the Beatles movies I've seen (well, I don't recall his role in
Magical Mystery Tour...

) IIRC he was kind of pivotal in
Hard Day's Night what with his goofy Irish uncle being catalytic and of course in
Help! he's the one who gets the ring (Ringo, get it?

) stuck on his finger.
It's very strange how hard it is to rent or buy any Beatles movies nowadays, or ever--I've only ever seen
Yellow Submarine for sale in any format, videotape or DVD. I have a copy of
Hard Day's Night on a disk--but it isn't a DVD, it's a CD from before DVDs were being sold, with the movie in Apple QuickTime format! I saw MMT once at a campus movie night screening back in the 1980s, and
Help! came on the TV once that I saw early in the 1990s, and that's it.
What's up with that? The same ownership snafus that prevented Carl Sagan from putting any Beatles songs on the Voyager records?
Perhaps the only reason Yellow Submarine is fairly readily available is that except for their little end of the movie cameo and their voices in the recorded songs, the Beatles were not actually the voice actors in the cartoon, so it wasn't really a proper Beatles movie at all. (And it didn't count toward completing their contractual obligation of a set number of movies either, much to their consternation).
As someone else pointed out, Ringo was the butt of most jokes and if they kept their old drummer the newbie dynamic that kept him in that place would not have been operating, so it could have been any of the other three just as likely to be stuck with that--probably Harrison from what I know of the history and dynamics of the group. Or Pete Best, maybe. If they picked a different replacement for Best, they'd probably be the low one in the pecking order, unless their personality and talent was such that it happened differently.
Really we'd need to know enough about why Best left in the first place and who was available at the time to replace him, and what they'd have been like. And then consider the possibility that in the wonderful world of entertaining arts, talent is a necessary but not sufficient condition for fame and quite conceivably (horrible as it is to contemplate) lightning would not strike and no one outside of the nightclub scene of Hamburg ever hears of them at all.
