Perhaps no band in the real world has had quite as much of an impact on the world of Doctor Who as the Beatles did. Both were of course fantastic products of England in the '60s, but some of the connections go far deeper than a contemporary complementary pop-culture acknowledgment.
Almost every single incarnation of the Doctor has in some way participated in or manipulated the history of the Fab Four. In fact, there is at least one universe where the band was still active up to the Live Aid concert in 1985. A companion of the Eighth Doctor from his post-TV movie book series, Fitz Kreiner, made it a point to collect alternative timeline Beatles releases, and could play "Let it Be" on the piano.
Presumably that means he has a copy of "Colliding Circles" and "Left is Right (And Right is Wrong)."
More to the point, the Beatles were once supposed to appear in Doctor Who for the First Doctor story, "The Chase." Set in 1996, an elderly version of the band would have been still active and still performing at the Festival of Ghana in a brief cameo appearance.
Beatles manager Brian Epstein ultimately vetoed the idea, but footage of the band performing "Ticket to Ride" on Top of the Pops in 1965 is still featured in the first episode. Vicki refers to them as classical music. A museum in Liverpool is dedicated to them in her time, and they're still well-regarded enough as composers to be pub-quiz fodder in the 42nd century.
There is something very strange about that footage. It is the only surviving film of that Beatles appearance on Top of the Pops. All other copies have been lost. Considering that William Hartnell and Patrick Troughton's runs as the First and Second Doctors are rife with missing episodes junked by the BBC, it's very ironic that the show is responsible for preserving an otherwise lost piece of Beatles television history.
Actually it's double ironic, because licensing issues had necessitated the removal of the clip from all non-UK or Australian DVD releases.
Had The Beatles still been active in 1996, it would have of course been extremely anachronistic not only because of the band's breakup in 1970, but because of John Lennon's murder by Mark David Chapman in 1980. Even Lennon's tragic death has ties to The Doctor in Kate Orman's novel The Left-Handed Hummingbird.
In Aztec times, a warrior named Huitzilin was exposed to radiation from a crashed Exxilon spaceship. The energy enabled him to become the personification of Huitzilopochtli, a god of war and sacrifice that would feed on death.
In 1969 Huitzilin sought to possess a random passerby to shoot John Lennon as he and the Bealtes performed songs from Let it Be on Apple studio roof unannounced... their last public concert together. Huitzilin's plan was to use the murder to inspire fear and panic in the world, but was stopped by the Seventh Doctor and Ace. The Seventh Doctor suspected that Huitzilin would influence the actions of Charles Manson in response.
In 1980, the Seventh Doctor would arrive just in time to see the Huitzilin-possessed Chapman finally gun down Lennon. The Fourth Doctor claims to have met Lennon at some point after his death in the novel Eye of Heaven. Lennon tells The Doctor, "Talent borrows, genius steals."
The Sixth Doctor played a show with the Beatles in Hamburg as part of a bucket list wish upon being told his life was coming to an end. He convinces Stuart Sutcliffe in 1961 to go home to his girlfriend, Astrid Kirchherr, for the night, and then offered to fill in for him.
The Tenth Doctor twice took companions to see them play at the Cavern in 1963, once with Martha Jones and once with Donna Noble. Donna made the mistake of asking them to sign a copy of the 1 compilation album... on CD. She then promises Lennon she won't put it on eBay, because sometimes Donna just didn't get it.
By far Lennon received the most attention by The Doctor, though Eleven once lamented that no one ever asked him to meet Ringo in the video game City of the Daleks. This is slightly strange when you consider that Ringo joined the band in 1962. Yet Pete Best is still clearly the drummer in 1963 when Ten and Donna attend, but is accurately Ringo for Eleven and Amy Pond.
This wouldn't be the last time Doctor Who muddled with the lineup of the Beatles. According to the novel The Devil Goblins from Neptune by Martin Day and Keith Topping, Paul McCartney did indeed leave the Beatles in 1970. In the Who timeline, though, this causes a divergence where he is replaced on bass by Klaus Voormann and Houston's own Billy Preston becomes a full-time Beatles member as well.
It's difficult to know which, if any, is the proper history of The Beatles in the Whoniverse. Their one supposed television appearance as older men in 1996 never actually happened, and should theoretically have been impossible after the Seventh Doctor watched Lennon cut down in New York. On the other hand, perhaps in another time, another dimension, history is as Eight and Fitz would have it and the Beatles still exist in some form or another.
After all, as we've proved before, The Doctor has certainly visited the lads in our universe. Why not the reverse?