As a little kid two of my main obsessions were music and accents (plus ça change...) and I remember watching "Yellow Submarine" when I was about five and thinking to myself how odd it was that the Beatles spoke with British accents but sang with American ones. Only years later did I realize that this initial impression wasn't entirely accurate.
In fact, the Beatles sing with what I call "Standard English Singing Pronunciation," which is a standard shared by most Anglophones, including Americans. Its salient (or perhaps typifying) feature is that it is non-rhotic. Indeed, just about the only popular American music that is rhotic is country music.
Now this is an odd situation, and it merits some consideration. There may be research on the topic, but since this isn't my field I don't need to know about it. Although an alarmingly high percentage of my readers are linguists, especially now after Ben Zimmer mentioned me in a post on Language Log, so I should perhaps think twice before flaunting my ignorance. Except ignorance is so fun to flaunt.*
In short, the situation is not that the Beatles sing with American pronunciation, but that Americans sing with British pronunciation, more or less. And yes, I know that there are rhotic British dialects and non-rhotic American ones. In fact, the latter will come to play in my made-up explanation. Firstly, though, it is well known from early recordings that for public occasions Americans (or at least Americans from a certain background) would adopt a very strange non-rhotic accent. It could be that singing, as a kind of public performance, also had this "fancy" pronunciation in America in some contexts. Perhaps then this became a convention that is retained to this day as a hallmark of musicality. On the other hand, not all music lends itself to conventions associated with prestige and formality; I think that in these cases (non-black) Americans sing non-rhotically in partial imitation of African American vernacular, which has, of course, had a profound effect on all American music. Thus non-rhotic singing has associations that are both formal and not formal. I think that now we are closer to understanding why country music, alone among styles of American music, adopted rhotic pronunciation, despite the number of country singers from the South, a region where there is fairly widespread derhotacization: in order to project an image of folksiness, country singers couldn't use a fancy non-rhotac pronunciation, nor were they eager, as white Southerners, to co-opt a hallmark of African American pronunciations. Indeed, the most famous African American country singer, Charley Pride, sings with a markedly rhotic pronunciation, even though he's from the Mississippi delta, where African Americans (but not whites) derhotacize.
On the other hand, a large number of Country singers are from fully rhotic Texas, and this is probably a factor too, as is the fact that the center of the industry is in rhotic Nashville. Hank Williams presents an interesting case; he is from southern Alabama and he sings with a strong rhotic pronunciation. According to a map on Wikipedia, which is based on Labov et al., the isogloss runs right through this area. I haven't been able to find any recording of Williams spekaing, though I'm sure they exist; it would be interesting to know whether he had a rhotic pronunciation or not.
So when, as a kid, I identified the Beatles' singing pronunciation as American, I wasn't entirely off the map. I believe I was also noticing the contrast between their Liverpudlian "Scouse" dialect and there more generic singing pronunciation. It seems that increasingly, though, U.K. musicians are starting to use distinctly local pronunciations. One prominent example is the "it" band of the moment, the Arctic Monkeys, from Sheffield, Yorkshire. So far they are following a now all-too-familiar pattern with British pop musicians: intense media hype followed by a nasty critical backlash in the equally nasty British music press. Others attack them for using too many localisms in their singing, which I think is sad. I must say, though, I don't really hear that many localisms, though I haven't listened to enough of their music to judge; I haven't heard them sing "tha" (thou) for instance, one of my favorite Yorkshire features, but I did catch an "owt" (something/anything). Here's a video of a live performance, and you can hear their dialect for yourself, although it is markedly less strong when they sing than the spoken bit at the beginning, my favorite part of which is when, in reference to all the flap in the press about them, the lead singer says "Don't believe the 'ype," pronouncing "the" as /ði/.
Speaking of videos, British musicians and their dialects, and backlash against heavily hyped bands, check out the following clip of the Beatles performing a very silly skit on a British T.V. show, I'm guessing in 1963. Notice their exaggerated Scouse dialect, but notice too the reaction of the crowd. In addition to the de rigueur screaming, there is a considerable amount of heckling, most of which consists of "shut up" and "go back to Liverpool." In a way it's comforting to think that even the Beatles had to face hecklers.