The V&A was my favorite museum on my trip to England earlier this year. You won't find a lot of woodturning, but you will find some, and you will probably find lots of inspiration for turning. I particularly liked the Asian galleries and the Islamic Middle East galleries. When I was there, several galleries were closed, but I still took the family back for a second visit before we left London.
I'd also recommend a visit to the British Museum. There you will find some ancient ceramic pots, alabaster vessels, carvings, etc. that should give any turner plenty of ideas for forms to try on (and for carving, off) the lathe. The collection is rather amazing (and I'm pretty sure that there was no admission charge — a welcome relief from the sticker shock of London food, lodging, etc.). You will also find the Rosetta stone and the Elgin marbles there — historically very significant, but not as interesting to me as other artifacts.
If you like woodcarving, check out some of the churches and look for the carving on pews — the pagan "green man" makes an occasional appearance.
Finally, I found some nice examples of treen in the antique shops on Portobello road.
Be sure to take a digital camera to make a record of things that might inspire you. (Most museums were fine with taking photos — some antique dealers were not.)
Have a great trip.
John