Usually, when it is not an American production, they are not too concerned with how well Jackie speaks English. Jackie can speak good English when he is talking to a crowd or an individual but that's because his words are not predetermined by words on a page. On his US movies, directors will film many takes of his lines to make him sound more fluent. Especially in the case of the Rush Hour movies and The Protector.
I think the English lines in his Chinese films are normally two takes at most.
Jackie dubbing his voice in his movies has always been good except for with Police Story 2013. I had a very hard time understanding what Jackie was saying. Watching it in the original language was fine but I had to rely on what other characters were saying to figure out what was going on in the dubbed version.