Granted you don't need a degree from Oxbridge or LSE, but I'd aim for top 15. You also need a first or a 2:1 most of the time.
You don't need a degree in finance or economics, but it would be useful as you are analysing the markets. And you need a summer internship in your second year, lots of the big banks recruit their summer interns after graduation.
The Economist - Have you got any work experience for banking?