This might be stating the blindingly obvious but London goes a bit nuts in December as tourist numbers soar (& the concept of 'work' seemingly escapes the indigenous population from around Dec 10th...)
Avoid Oxford Street/Regent Street, especially if you're there on a weekend.
The idea about taking the tube to St Paul's is a good one. Another option for some views is the Thames Clipper water bus (note not the tourist cruise). If you can factor this in correctly then you get to see Tower Bridge, The Tower of London, Houses of Parliament plus all the new skyscrapers from the river.
Make sure you allow long enough to get back to the airport. It's 1hr on the tube and that's before you allow for delays so by 3pm you need to be gone.