I am not British born but lived in this country for 13 years and practised as GP for the last 7 years+ 3 years of being GP trainee. Not at any point of my entire life ( including professional part of it) I felt discriminated. Not a single patient ever commented on me being foreigner in a negative manner. I have had British citizenship for 10 years now and very proud to be British. I’m not even slightest concerned about being intimidated by any comments on my nationality. I really couldn’t care less as I am more patriotic and contribute to this country more than some of the people who were born here. So i would not even pay attention to any comments related to my nationality.
Some teething problems are always expected in any transition. Your story is a perfect example that people who are highly educated, professional and make a massive contribution to this country do not need to worry about brexit. Some communication might be needed with Home Office etc but I have no doubt that the UK will treat hard working professionals with respect and things will settle after a short while.