Waterloo is a big, wide bridge and connects to major routes at both ends. I can't help thinking that closing it would simply move the problems elsewhere rather than solving them. The junction at the north end of Waterloo Bridge needs work - it has already been improved somewhat in recent years with changed priorities and phasing of the lights to benefit cyclists, but there's still plenty of room for further improvement.
The ones I would close would be Southwark (but keep Blackfriars and London open either side of it), and Tower, which is too narrow and permanently clogged.
Looking at Google Maps, it seems they have already made significant improvements (since I last used it) to CS7 south of the river between Southwark Bridge and E&C, which is good. Closing Southwark Bridge instead of London Bridge would be a further improvement.
I would also reconfigure the north end of Southwark Bridge so it goes over Lower Thames St rather than connecting to it. But that's wishful thinking.