It'd be harder to keep McCartney from knowing Harrison than the rest, they were friends before McCartney met Lennon. That doesn't mean they were destined to be in the same band or anything, just that I'm not sure how to achieve the idea that they never cross paths at all. Although this is really just-no Beatles and I'm splitting hairs. It's really easy to imagine a world where they don't make it. McCartney would probably have a musical career of some sort anyway, but Lennon probably ends up like his father eventually, though I like the idea of John Lennon-British humorist/writer/cartoonist personally.
Speculating on what bands would take the Beatles place in history is jumping the gun a bit. A lot went right for the Beatles to make it in America, so while another group could make the leap, it isn't inevitable-or at least it isn't inevitable in the same time frame.
Without the Beatles, we wouldn't have the Rolling Stones. Sure, there might be a blues cover band featuring Brian Jones, Mick Jagger, Keith Richards, etc. but they wouldn't be the Rolling Stones we know. Their image was crafted and cultivated by Andrew Oldham to make them the anti-Beatles-and Mick Jagger's actual personality at least didn't actually fit that image very well in reality so I'm not sure that's the sort of thing that would have emerged organically anyway. Without the example of Lennon and McCartney, Jagger and Richards might not start writing their own material, at least not for a while longer. The Rolling Stones may not even be signed to Decca, as they were on the suggestion of one of the Beatles, I think it was Lennon but I could be wrong.
Led Zepplin is years away, so there's no telling if it will exist. We don't even know what the Yardbirds will be up to, let alone if Jimmy Page will end up a member.
Other groups are less dependent where the U.K. is concerned than the Rolling Stones, but making it in America was a difficult proposition.
I think that without the Beatles, the British Invasion could itself be prevented or delayed such that music in America in for the majority of the 1960's is dominated by folk and Motown rather than some alternate British act, and Rock and Roll becomes an artifact of the 1950's.
