I'm American (but most definitely not a Yankee! *grins*), so please forgive me if I end up saying something utterly foolish in my ignorance of British politics.
It seems to me that Mr. Cook was trying to make the point that Britain has a long history of immigration and will not be led to ruin by it now. He doesn't say that there is no such thing as "British", merely that "British" is not a race. It's not, nor are American, French, Italian, or Greek.
"Race" in a biological sense is meaningless. We're all the same species--"Negroid" and "Caucasoid" make no more difference to anything than "blue-eyed people" or "redheads" do to a biologist. Race is merely another physical characteristic.