The Beatles and every other Sixties British band lucked out when National Service was abolished on the cusp of calling up all the founding members of the "British Invasion." Had National Service remained in effect, all of those proto-bands would have been torn asunder before making their mark, almost certainly never to reform.
Had the Beatles been conscripted, what regiments might they have served in?
It's fun to guess. It would be even more fun for some clever PhotoShop expert to create some hypothetical portraits of the military Sgt. Pepper's band. We have a pretty good idea of what John Lennon might have looked like thanks to his role as Musketeer Gripweed in the 1967 movie "How I Won the War," altho the WWII battledress was disappearing by 1960. I like to think of him in the Liverpool Scottish, given his affection for Scotland and bagpipes, in a Forbes tartan kilt and glengarry. But this being a TA regiment at the time, it probably wouldn't have been an option for a conscript. So maybe he winds up in the King's Regiment (Manchester and Liverpool) or perhaps Lancashire Fusiliers, local units. Same for Paul McCartney when his time comes.
George Harrison I see perhaps going to the Loyal Regiment (North Lancashire) (successor to the Loyal North Lancashire Regiment, in which his grandfather Henry served and died in the First World War (in 1915, on the first day of the battle of Loos).
Richard Starkey? Hmm… too small for the infantry, I'd say he winds up in the Army Catering Corps or some similar support unit, washing dishes, peeling spuds…. or goes to the music school and winds up playing the snare in a marching band!