That's a big question. The US has a pretty big influence on most European countries, but it's not like they have complete control over them. And probably subject of crypto trading is not of any major strategic importance, so maybe they won't be expanding their crypto crack-down outside of the US.
If I remember correctly, the most recent opinions of EU on crypto were not very friendly, and Binance had to block certain features for its UK customers (as well as well to stop supporting GBP deposits/withdrawals), so I'm not very optimistic here, but wish Binance all the best.
I don't fully agree with you pawel7777: although as western economic powers it is logical that they are similar in many aspects, and North-American cultural influence over Europe is undeniable (but let's don't forget that vice versa also), there are many economic, social and political conflicting interests between them, and that's the reason why the UK left the EU (Brexited), because their interests fit better with the Commonwealth's than with the Union's.
So I think that you are mixing things up, and if Binance's way is eventually blocked in Europe I don't think it will be due to the influence of the US.
I'm very curious about the answer to zasad@'s question too, but we'll have to be patient until time tells and make decisions accordingly.