I had a mortgage with NatWest for a while as a US citizen, they had zero issue with it, but didn't have a bank account.
Other banks (or bank-like institutions) I've dealt with and had no issues include Nationwide, Monzo, Starling, Paragon, Barclays, Halifax and Wise.
The only bank that's turned me down was Zopa.
Investment companies is where the list gets an awful lot shorter. Hargreaves Lansdown, Interactive Brokers, and some types of accounts at A.J. Bell is pretty much the whole list (outside workplace pensions, where I don't think they get a lot of choice).