You would be fine in all of those places. They’re all major hubs in some right.
The institutions attract many BAME students in each of those universities listed.
You should be fine, dw, there’s some level of racism’s everywhere and truth be told, you’ll experience it wherever you go as long as it is not a small village type of scenario.
The point about educated people and being less racist is definitely pretty accurate or reliable. These are all cities with a good level of education (compared to the rest of the UK).
I’ve personally been in Manchester and Liverpool (lived both places) and I was fine in each.