I haven't seen that "Yesterday" movie, so I can't comment on that. But as youth culture generally was a thing emerging in the 1950s, I'd say that at least musically, developments would be similar up until the mid-1960s. Beat probably also develops like in OTL, maybe with The Searchers becoming the biggest beat group in the UK. The musical development in the UK would be roughly similar, albeit with a smaller cultural impact on a wider and global scale. You'd still have the youth cultural craze, you'd still have bands like the Stones, Animals, Kinks, Hollies, The Who, Manfred Mann et al, but without the Beatles kicking in the door, they would be nothing more than just music groups in the average person's mindset. Maybe it'd be more like OTL punk - a huge cultural movement with distinct music, clothing styles and attitude, but with the average man or woman just thinking "those crazy young kids, they just make noise".
In America, pre-British invasion styles like surf, rock'n'roll or blues would hold out a bit longer than OTL, and without the Beatles as the driving force behind the British invasion, the US charts would be dominated by Tamla-Motown, garage rock and folk.
Two distinct things come into my mind when it comes to the early musical impact of the Beatles:
1) At that time it was quite unusual for bands to write their own songs and perform and release them. In America, this was changed by the advent of singer-songwriters like Dylan, but in the UK it was the Beatles that paved the way for bands writing most of their own material. So I guess without the Beatles, the songwriter/performer-division would remain for a longer period of time - possibly with singer-songwriters being seen as a bit strange.
2) In the Fifties and early Sixties, bands used to have names like "Bill Haley and the Comets", with a band leader and his backing band. That changed with the Beatles becoming successful. So maybe we'd see band names like "Mick Jagger and the Rolling Stones" or "Eric Byrdon and the Animals".
I guess we'd see more divergence from the mid-1960s onwards, maybe with newly forming bands being much more influence by the Stones, the Byrds and the Who, therefore setting the template for blues rock, folk and mod as the sound for the pop charts. But honestly, it already gets too butterflied at that point.