I doubt there is really a completely neutral British English accent. There almost always small give aways in an individual's speech which carry influences of both region and class. Having been born and brought up in Solihull, a south-eastern suburb of Birmingham, I grew up able to speak with a West Midlands accent and alternatively with a supposedly neutral British accent. The latter was acquired by being sent to a posh boarding school by my parents, who were determined I should not speak with a Brummie accent.
I believe the West Midlands accent uses more vowel sounds which differ from the neutral (or BBC, or "Oxford" or Queen's English) pronunciation than any other regional accent. I think this is why West Midlands (or Birmingham or Brummie) is so despised. Personally I love it; it's the linguistic equivalent of a delicious mature Blue Stilton cheese.
The inhabitants of Solihull are somewhat obsessed with trying not to sound Brummie. They will even go to the extreme of saying in their obviously West Midland tones: "We're nothing to do with Birmingham, you know. We come from a little village near Stratford upon Avon". Their feigned association with Will Shakespeare is comic. I have no doubt that the playwrite/poet/actor spoke with a version of the West Midlands accent used in his day and, like any good actor, could probably mimic any accent needed for whatever part he was playing at the time.
Let's not get too serious about it.