For your backup idea you must consider that its Olympics time so crowds and prices will be up accordingly.
If you are doing the IWM then make the most of London and do the Maritime museum as well, likewise with Gosport do the other side of the harbour for its museums too.
As for keeping costs down, go to the north it might be grim but its a lot cheaper
But seriously, even in London B&Bs are fairly cheap and there is no shortage of reasonable hotels either, plus with their London transport system it is very easy to get around.
Also on a costs thing, make sure you get all the best travel discount cards, they may seem like a silly expence on vacation but in reality they pay for themselves very quickly(sometimes in just one train journey). Another thing on public transport over there is to buy tickets in advance as you can make huge savings.
Oh and despite what the Brits will tell you their public transport isn't bad and in London it is excellent.