Some do, some don't. Lily Allen is the first singer that comes to mind when I think of an English singer that still keeps their accent. John Lydon still keeps his accent when he sings, just as he did back in the days of The Sex Pistols. The Clash is an english punk rock band and I don't recall Joe Strummer sounding English when he sang. I think it has something to do with what/how they sing and whether or not they emphasize some vowels.
It may simply be that when we sing we enunciate our words more carefully and create a kind of lingua-franca of song so that we all sound a little more like each other than we do when speaking.
When I was calling square dances Daizy warned me when I occasionally picked up an American accent but that was slightly different. I learned the dances by listening to American callers and would sometimes unconsciously imitate them.