It's true that there is some amount of "hate speech" of the form "let's kill all the $(DEMOGRAPHIC). Hitler had the right idea!" That speech is being censored in the UK.
There's also a lot of legitimate debate being censored/silenced in the UK. "Let's put an end to fake asylum." "Let's stop allowing chain migration." "Some of these people are never going to assimilate even after multiple generations, we should stop repeating the lie that they will." I don't think that's "hate speech," that's speech advocating immigration policy.
Most of the people claiming to want to end "hate speech" point to examples of the first case but most of the actual censorship is of the second case.