Loads of people have done it (what jonnijon was saying). You have to pay a fee though. My husband has done it thru the hotel reception where we once stayed and we only have to wait a day at an extra fee.
Regarding Brexit, visa processing for non-EU residents won't be affected in my opinion. It's already hard enough as many would say.Brits taking non-EU spouse/partner here are mainly scrutinised more compared to the ones from EU countries. You need to have plenty of money
to be able to nowadays.