Seriously though, Britishness (in the genetic sense) is a fundamental part of James Bond's character. He isn't just British by citizenship, he's emblematic of Britain.
Firstly, maybe in the 60s, the idea of a black person being "British" would be scoffed at; today, with 3 or 4 generations of genetics here() and black people representing Britain at many levels, I don't think it would be a problem for most.
Secondly, 007 has always been an outsider and not a member of the Establishment - that's part of why he does things differently to the other 00s (although another commenter says that's not part of the books, only the films - I've not read them so can't say).
() ignoring the fact that there have been significant African populations in cities such as Liverpool and Bristol since Elizabethan times
