One problem with the "regular" English accent is that it mostly consists of short vowel sounds. This reaches positively ludicrous levels in Lancashire (where I'm from), where people literally don't finish pronouncing the ends of words or sentences - it just kind of tails off about 90% of the way through. This is a tendency which took me quite a while to fix when I started singing lessons!
The problem is though that if you simply elongate your vowels, your accent becomes very "upper clarsse", because that's the major characteristic of the upper-class accent. This might be fine for classical, but it doesn't work for rock or pop, nor folk either. So I think that's a major factor for why British people often end up with a slight American accent when they sing - the vowels are slightly shifted in a way which allows for a more natural-feeling delivery, but this has the effect of changing the accent. Singing "laave" for "love" is a perfect example of this.
It might also be worth noting that it's thought the American accent historically *was* the British accent. But the Americans kept the same accent (and words like "catercorner"), whilst the native British-English speakers changed how they spoke and lost some words from their vocabulary.