When you say "British" / "European", do you specifically mean "White?" (Anglo Saxon)
Either way, to answer your question, some will, some won't. However, I don't think this board is the best place to get an accurate answer, as there are a lot of desperate people here with no experience with women, who would literally date anyone who would look at them.
So yeah, you'll probably get a lot of "yes" answers here... but how much it reflects reality (and whether you would date the guys in question) is a completely different matter.