http://www.talkingcities.co.uk/london_pages/shopping_music.htm

Forgive inability to make this a blue link. Brain turns to mush when I reach number three on any instructions.

But this site will give you addresses of heaps of secondhand record stores in London....and the ones listed are all pretty big. Quite a few have a number of branches all over London, but if you are doing the British Museum you will be close to loads of them.
They stock just about anything and everything, rarities and all. Every genre catered for.

Just remember your A to Z and a travel card. Have a good day!