You know the situation very well
Durham's words have a long history, but finance is really rich. The graduation thesis is only 8000 words, which is basically a major of making money.
Birmingham, business is ok, and the school is well-known, and sometimes it can rank among the top in the world 100. The school is very big and the city is very big.
Personally, I think you'd better go to Birmingham, as long as you arrange your study time reasonably. There are not many postgraduate courses in Britain. Some people say that one year is tight, but there are few courses, all of which are theoretical.
You can send me a message if you have any questions.