In the UK black people have to contend with some racism, but generally they are pretty much integrated in many ways. The focus of British racists has shifted over time towards Muslims and east Europeans.
Thing is, that although the Brits were very much involved in the slave trade back in the imperial period, the slaves went mainly to colonies like America or the West Indies. There were very few slaves here. Immigration of Blacks, mainly from the Caribbean started after WWII when Britain had a labour shortage and many were encouraged to come over.
British racism is constructed differently from the US variety partly for that reason. Backs were looked down on historically, but from a distance.
There's the usual racist fear of anyone different, but nothing like you've had in America.
When Britain was an imperial power, they had to tell themselves that blacks and other non-Europeans were inferior in order to justify to themselves colonizing their lands. I think that's really one of the main the bases of British racism even now with regard to blacks. It's slightly different in the case of the other immigrants I mentioned.
I wouldn't say that blacks don't experience racism here or that they are fully accepted by everyone, but it's not as bad as the USA to judge from what I see in the media, which as I've never been to America is the only source to asses things.