Do the people in Reniassance festivals occurring in Britain also speak with faked British accents, or do they use faked French/Italian accents instead?

Renaissance fairs are an American thing. They don’t exist in the UK.

I don't know the UK, so I might be wrong, but it's not an US only thing (beside the naming).

I've seen various form of medieval markets/festivals on several continental Europe countries, and I don't see why it wouldn't cross the channel. Even though, Historical re-enactment is a niche hobby, it's not that uncommon to find group focused on the a given historical period caring a lot about the details, no matter if they re-enact the Viking, The crusade, or Napoleon's army (If you visit Belgium during the Waterloo battle anniversary, you can see these groups re-enacting the battle). LARP is also a thing in the whole western Europe, it's way less realistic as we do have elves and orcs, the biggest worldwide runs in Germany Trailer and UK has also some big ones and a lot of smaller ones.

So I'm sorry to disagree with your claim