To be fair I don’t think it’s just the UK where this is the case. In fact I’d argue the UK is one of the most accepting countries.
I spent some time in S.Korea, there your looks absolutely do decide whether you are Korean or not.
No matter how many generations you’ve lived there.
It may be different for the younger generation, but when I was there a black/white/japanese etc. person saying they were Korean wouldn’t be believed.
I think that is true in many countries.