I'm doing a PhD in English which is my first language, so not in the same boat at all, but just wanted to give something from that perspective. People in England are so used to hearing non-native speakers speak English that it often doesn't register when people don't speak perfectly. I have been in situations with my non-native speaking English friends where they all turn to me to adjudicate an error in someone's English, where they aren't sure or in agreement and I won't have even noticed. I mention this as I think it's hard when speaking a foreign language as you think native speakers hear all the mistakes, and I just wanted to say that much of the time we don't Certainly not if we're used to hearing non-native-speaker-English, which in unis people are.

And for what it's worth, the time when I realised I could really speak German was when I thought in German rather than thinking in English and then translating. So immersion, and embracing English all the time is imho the way to go.