Joe's cafe next to Borough Tube (not too far to walk from London Birdge) is brilliant in my view - clean north american dining experience with excellent menu, including some healthy favourites alongside some less healthy dishes! Its open Sunday too.
Saturday of course also offers up Borough Market for a range of "on the hoof" eats.
Both Pont de la Tour and Butlers Wharf Chop House near Tower Bridge have a fixed price lunch menu of £12-£15 for two courses per person on the weekends. It's a great opportunity to eat at two top-notch restaurants without breaking the bank.