My wife and i did exactly that this past thanksgiving, leaving IAD on the Friday morning before Thanksgiving week and returning to the US on the day after Thanksgiving.
The weather was windbreaker chilly, grey and overcast with brief intervals of sun. Usually the rain was drizzle, but there were a few hours of downpours. However, London was no problem at all. (We were outside more in bath and Liverpool, so the weather was more of a factor). Any long trip was done by Underground, and brief walks down city streets were actually a relief....the English tended to overheat their interior spaces, so we would have to strip down to shirt sleeves immediately after entering a building.
There are so many museums to see that you could use up the entire time you have doing museums and never do anything else. We even did a nighttime Jack the Ripper walking tour and loved it. So no, don't worry about the weather. Bring layers of clothes and a long waterproof poncho with hood and perhaps a spare pair of shoes (or waterproof ones)...you'll be great.