We were there for Christmas Week last year. I cannot overestimate the caution about things closing down--- Christmas eve is not the night to do last minute shopping. We went to a 2pm matinee in the theater district on Christmas eve and by the time it let out at four, things were definitely slowing down. I knew it would be slowing down, but this was significantly more curtailed activity than I expected in a major city.
Aside from a play on Christmas Eve, the walking tour group "London Walks" gives 1-2 walks on Christmas Eve and possibly Christmas day. Because there are so very few things to do on Christmas day, these tend to be very crowded and remember you need to be able to walk to the meeting place and back because there will be no transportation at all on Christmas Day. It becomes critical to think about transportation on Christmas Eve and Christmas day because if you are doing something out and about on Christmas eve or Christmas day, you need to make sure it is within walking distance of your lodgings.
Even restaurants starts shutting down about 3 o'clock. On Christmas Eve. We found a pub for dinner about 5and by 7 pm even the busiest London streets were nearly deserted. A few buses and tubes and a few taxis were still working, but it was slim pickings. I was very surprised that by 8 o'clock at night, nearly EVERYTHING had shut down.
My experience was that on Christmas day everything was closed with the exception of a few, tiny family-owned mini markets. Because traffic is so limited on Christmas Day, it is fun to rent one of the many bicycles on the sides of the roads and take a bike ride. This is popular and in fact we had difficulty finding two bikes on Christmas Day. We persisted and it was lots of fun to sight see in all the tourist areas with no traffic out and about.
There are hotels and pubs that have special meals on Christmas, as well as some ( but not all ) Indian and Chinese restaurants. You must reserve ahead of time---be aware it can be very expensive. We were a family of four and were unwilling to spend a total of $300 --especially because we have two very picky children! Happily, we stayed in a wonderful apartment and cooked a simple holiday meal for a fraction of the price.
Again, if you are going to be there at Christmas time, this might influence what area of London you lodge in--stay so that you are within walking distance of things to do on Christmas Eve and Christmas day.
PS Re Christmas weather: We live in San Diego. I brought 4 mini umbrellas 6000 miles to London, and 6000 miles back and in the 10 days we were there, it never rained. It was between 35 and 40° most of the time. No snow, no ice. We were likely exceptionally lucky but it was certainly lovely touring weather. Enjoy!