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.
Idris Elbra is as British as they come.
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