Christmas Day - as noted above - will be incredibly dead. Depending on where you are staying the period from there through to the New Year can either be deserted - City, lots of residential areas - or like the West End heaving with tourist hordes. Plenty of the good restaurants and bars also close during this period leaving you with a less than inspiring selection of dining places.
Haven't spent NY in London for at least 12 years and no plans to start either.