Try Supreme in Kennington Road 020 7735 5819 just by the Imperial War museum, at the end of Brook Drive - they are excellent, no nonsense, old school enthusiasts, who deserve the support of local motorcyclists.
I know nothing about motorcycles but do know that the motorcycle repair shop referred to above, closed for good before Christmas. I'll have a look in the window when I next go past to see if they recommend anywhere else.
Harriet I would strongly second the Chas Bikes recommendation. Having tried a few I can assure you they are the most reliable and very reasonably priced, should any work need doing to pass your bike. No association etc just a satisfied customer of some years.
I popped into Chasbikes on Sat to get a new battery. A low-margin, fairly cheap, but quite fiddly thing to sell (to match the OEM battery to a replacement). I didn't even want them to fit it. But they were really helpful and took time to make sure I got what I needed.
I'd never been there before, but I'd go there again. They have a website. (No connection with them, etc.)