Very cool question. Here's my take on an answer (the Bukowski version):
Singing does weird things to a language. Specifically here, it minimizes or even erases many of the sound differences between American and British English. There are some technical reasons for this, but I'll leave them out for now. The punchline, though, is that it's not necessarily the case that British singers are trying to sound American. Rather, because singing makes it harder to hear distinctively British speech features, a singer's accent will sound more "American" to Americans (and, conversely, an American singer will in some circumstances sound more "British" to Brits).
Herman's Hermits is an interesting exception here. They really seem to go out of their way to drive home just how incredibly British they are, and I suspect that if we were to bust out the spectrograph on the lead singer we'd actually find that the contrasts are more exaggerated when he sings than when he talks. If anything's intentional, it's actually sounding
British, not American.
So, basically, the natural course of things would be for British singing and American singing to converge on something that sounds very similar, but musicians on both sides of the pond can exaggerate the differences for stylistic effect. This even happens in different songs by the same singer. Compare "When I'm Sixty-Four" to "Helter Skelter", for example.