The conventions of popular song in the English language since the mid-twentieth century have led to the dominance of what's been called a mid-Atlantic accent. The general belief is that the prevalence of extended, possibly distorted vowel sounds aren't really compatible with British accents as spoken, even by singers. The exceptions are very noticeable and they do exist: Billy Bragg, the early recordings of David Bowie, and Anthony Newley, for example.
See this blog by David Crystal:
Several of the main identifying features of a regional accent tend to disappear when singing - the intonation (obviously, as a melody replaces it), the speech rhythm, and vowel length (for many syllables are elongated). Vowel quality is also often affected, especially in classical singing, where vowels are articulated with greater openness than in everyday speech.
In the Classical world, the situation is similar. From Singing and Communicating in English: a Singer's Guide to English Diction by Kathryn LaBouff (p.242):
For singers today, knowledge and fluency in the Mid-Atlantic dialect is a very useful skill. In North America, it is often the requested pronunciation by many conductors and directors for vocal works that are not specifically of North American origin. Oratorio and European opera in English translation are frequently presented in Mid-Atlantic rather than RP or AS.