According to Wikipedia (and we all know that they are THE final word), there are 240. There are 24 that specialize in health and medicine alone.
Like the the Anaesthesia Heritage Centre, which is run by the Association of Anaesthesists of Great Britain and Ireland. It is in Portland Place and is open to the general public. The centre is open Monday to Friday from 9 to 5, but is closed on Wednesday. There is no charge, but appointments are recommended. It comprises a museum with a collection of objects dating back to 1774, the Association's own archives, which date back to 1932, and a reference library.
I couldn't find any information about the museum on the Association's website, but here is the link for the page on Wikipedia.
Then there's the British Dental Association Museum, which has over 20,000 items. The museum is located at 64 Wimpole Street, London, W1G 8YS and the website is here.
After seeing some of the archaic dental tools, like the toothkeys above, you might need some psychoanalysis, which you can find at the Freud Museum. Located at 20 Maresfield Gardens in Hampstead (below), the museum was the home of Sigmund Freud and his family when they escaped Nazi annexation of Austria in 1938, and was the family home until 1982 when daughter Anna passed away. Freud's study is the centerpiece of the museum, complete with couch (top photo).
For a list of all of the London Health & Science Museums, with links to their official websites, go here.