Where I live in the eastern USA, it is more common to say BRUSH UP ON, but the Cambridge Dictionary says that either form is correct: Definition -- To brush up (on): to improve your knowledge of something already learned but partly forgotten: I thought I'd brush up (on) my French before going to Paris. Perhaps "brush up" without the "on" is more typical of British English?

Both sound OK to me. brush up ___ brush up on ___ brush ___ up

Both sound ok to me. Brush up on is possibly more common, though.

I need to brush up on my English

brush up is phrasal verb and you can a preposition within it which the proper preposition is (on).