Probably would have gone to yet another record label.
The idea of The Beatles on Pye fills me with woe mainly for the hideous way the records would have sounded *shudders*
What labels were there, besides Pye, left to try in Britain? I've heard the Beatles were turned down by every label until they managed to get an audition for Parlophone and were picked up by them. I believe Parlophone/EMI and Decca were the only labels to let them audition for them. I wonder if maybe they'd try to get an audition with one of the labels that previously did not let them, using the Decca tapes and the EMI/Parlophone tapes to shop the group around. I wonder at the success of that.
I don't feel the Beatles would never get picked up. Not by a long shot. But it would be delayed, which would certainly have an effect on their music and their history and their affect on culture and history. Not to mention how a different label would affect their development, output, public relations and reactions, etc, which would be different and be rather major in (alternate) music history.
There is something to be said for the Beatles not getting picked up by Decca, because that gave them time to develop, and that time between the Decca audition and the one for EMI was limited but was very important because they were a much better band even those relatively few months down the road than they were at Decca, with more material, which we've come to know off their earliest singles and albums and couldn't imagine the Beatles without. (Whether they'd end up doing those with Decca anyway is an argument to be had, but I'm going with this train of thought). Such leads me to think that, had the Beatles been rejected by EMI/Parlophone, that would give them even more time to develop, so that when they were signed finally, they would come out of the starting gate extremely strong (stronger than the OTL) with a wealth of recordable material.
There is a criticism to be had of the earliest Beatles records in Britain with the fact that there were a litany of covers that really could have been replaced with something better. When I say that, I don't mean the covers in general, since doing covers was a standard, and there are many well done by the Beatles. What I mean is that "A Taste of Honey" really begs the question of why it's on there, and a wondering of all the better songs the Beatles could have covered and put in it's place, and the fact that original material could have been put there. Covers aren't the problem; it's covers that could have been replaced with stronger cover versions of songs, or original material.
That is the difference between the first two Beatles album, and the butchered track list of "Meet the Beatles!" in America. "Meet the Beatles!" is probably a stronger album than "Please Please Me" and "With the Beatles" because it combined the two, cut out many covers (because they felt Americans wouldn't care about the Beatles versions of them), and added singles. There is an argument to be made that the strength of "Meet the Beatles!" because it was butchered made a strong spark for the major Beatlemania that swept the United States and the Americas. You can certainly decry Capitol for butchering, and for putting singles on the album (which the Beatles didn't like because they felt it cheated the customers) but it did make for a very strong "first" album.
Taking that into account, had the Beatles remained in the creative womb longer, they could have come out of the gate with more and stronger material to really suckpunch the public on the earlier singles and albums of this alternate Beatles. Such a thing could also mean that a major American distributor picks up the Beatles shortly thereafter, America picks up Beatlemania not long after they hit the British charts. America was about a year late to picking up on the Beatles (1964 compared to 1963) so while Britain would be delayed, America could be right on schedule or right around the same time, give or take.
From there though, the arguments can be made about the lack of George Martin hindering their growth and capabilities to do new things and experiment, and a label and/or producer less open to the Beatles doing as they please forcing them to do the same things over and over and not allowing them to grow. Those would be very bad things.
