I don't think there's any chance for a British invasion, and I'm quite knowledgeable on the subject.
The British Invasion didn't happen because America loved British music (i.e. an amalgamation of American music through a British lens), it happened because America loved the Beatles.
And America loved the Beatles after Britain did.
The British Invasion occurred after British studio moguls began to profit from the unlikely success of the developing Mersey Sound, of which The Beatles were the only real vanguard. Then, and only then, did Beatlemania in Britain transfer to the British Invasion in the U.S.
And even that was an enormous long shot.
Before the Beatles were even widely played in America, George Harrison visited his sister Louise in Benton, IL (much of my family's home region, Little Egypt) and was relieved to find no one knew who he was, really.
It was only months later that the Invasion happened.
So, no Beatles, no chance of Liverpool (a backwater) becoming nationally known in Britain as this hip music town and therefore, no momentum for British rock/pop artists to do anything at all in the States.
And no British Invasion means no Garage movement, which means with all the first generation rockers dead, Christian, in jail, in the Army, or married to their cousins, the pop charts will continue to be filled with milky white, harmless rock imitations until someone gains success rocking the boat - which could take a while. The butterflies here are unbelievable and tragic regardless of how much you like the Beatles or don't.
YOUR music is butterflied away too. Deal with it.
But that's not even the best part:
No Beatles means no one takes the risk on artists who write their own songs, i.e. expect a lot of unoriginal artists because it's safe and profitable.
It also means no experimentation in the studio. I'm not even talking about the trippy stuff, I'm talking: "Play your bass part and get out. You have no say on loud it is, how it's produced, etc."
Producers will continue to hold the power that the record companies haven't already claimed, leaving the artists as pretty faces with little to no talent.
Black music, however, takes a big boost initially.
Motown won't get as easily synthesized after Barret Strong's hit in 1960, leaving Black musicians who play soul as the only way for white kids to hear it.
This is good and bad:
It's good because, like rock and roll, there is once again a movement that blacks can embrace and whites can sneak off and listen to.
IOTL, the last real black and white union in music was fifties rock and roll (en masse), and after whites took over rock completely during the British Invasion, blacks predominantly listened to black music (from motown to hip hop today) and whites predominantly listened to white music (from British rock to whatever it is whites predominantly listen to now. Country? The Pop charts? Shudder...).
No Beatles means the next wave of rock has a good opportunity to be as biracial as original rock.
Of course, it's bad because there won't be near as much musical competition. A black artist won't have the incentive to do black music better than whites (believe it or not, the songwriters at Motown spent years trying to out-Rubber Soul the Beatles) because whites simply won't play it to begin with. Problem solved on their end, but as listeners, both racial cultures will play it a lot safer.
I could see funk not existing because the desire and ability to experiment with soul won't exist.
No Beatles may mean someone else comes along to replace their role, but it may not be for decades, and in the meantime, with less competition and artists unable to be directly involved in their work, it's going to suck.
In short: Good thing there was Beatles, even if you don't like 'em.
EDIT: By the way, The Stone only got signed because George Harrison told Decca to sign them at a talent showcase where he was the judge. Still smarting from their passing over of the Beatles two or three years before, the company jumped on it. If this doesn't happen then the Stones (who were on the verge of giving up as one single band) would have split between Brian Jones (who had talent) and Mick Jagger (who didn't, but had swagger). Keith would be torn between the two, but as Bill would have gone with Brian, so too could Keith, leaving Mick perhaps fronting another Crawdaddy Club favorite, like the Yardbirds or later the Small Faces.
They are unrelated bands, certainly, but no Beatles basically also means no Stones. Sorry.