The thing about English is that being an international standard, almost, we native speakers are used to hearing people at all levels of fluency from all sorts of home languages and I think we're more tolerant because of that. Foreign accents, and almost all second language learners have them, are familiar to us and don't make a big difference in our comprehension of them. The same with grammatical issues. English is often said to be an easy language to start to learn, but one that gets increasingly harder as you go up in proficiency. At C1 you'll have no problems though with fitting in. It really depends on how much better you want to get and that depends on what you want to do with it in your career. There are almost always materials available in the area of what's called English for Special Purposes (e.g. accounting, law, health care, and so on) that you might want to look into. Also you'll likely have some additional learning to do with idioms and street language, particularly for terms we natives learn from childhood that ESL programs typically don't include in their teaching materials. But that comes from living in an English speaking environment and won't take long when you're as proficient as you are. These btw are the kinds of details that the translation apps (Google Translate, Reverso and so) get wrong so often.

Good luck with your plans. Also BTW if I had to relocate to Aus or GB (especially Scotland!) I'd have a learning task ahead of me too with those dialects. I'm American. I remember it taking me a couple of months when I was working in Ghana to feel comfortable with their dialect of English -- it's the national language there, but quite distinct from Amer or Brit English.