We are going to be in London for a few days before and after Christmas and are looking for suggested Christmas activities and a recommendation for Christmas dinner.
The main warning is that, as alluded to, there is reduced transport over the Christmas period. Christmas Eve: transport ending earlier than usual Christmas Day: NO TRANSPORT except for limited service to major airports. Boxing Day: Bus and tube running with limited service, no trains except to Heathrow. So either eat Christmas dinner at the hotel, or book somewhere you can walk to. In the run-up to Christmas, shopping areas will be very busy. In the last couple of years there have been German-style Christmas markets around the Royal Festival Hall and in Hyde Park but they aren't all that "British".
Trafalgar Square with its huge Norway spruce all lighted with various groups of carolers singing nightly is one of our favorites. Also, check at the Royal Albert Hall for Christmas programs that include all of the favorite songs of the season. Don't overlook St. Pauls for Christmas Eve and Westminster Abbey for Christmas day or vice versa. Stunning services at both places even if you are not church people. There are many Christmas markets some along the Thames near the London Eye. Be sure and view the festive window decorations at Harrod's. We ate Christmas dinner at one of the many hotels that offer them. We stayed near Paddington Station so we ate at the Paddington Hilton. You do know that there is scant transportation Dec 24-26? It was one of our best Christmases ever.
Ice skating at the Tower of London and other places. http://www.timeout.com/london/things-to-do/ice-skating-in-london Also at Somerset House on the Strand. http://www.somersethouse.org.uk/ice-rink
You will love it! I spent Christmas 2011 in London and really enjoyed it. Like others have mentioned, be aware that there will be limited public transit around Christmas and Boxing Day - so you might hope to have a well-located apartment or hotel. Also note that most of the museums are closed until after Boxing Day. I'll second recommendations for Trafalgar Square and its huge tree, and Westminster Abbey. I went to the First Eucharist of Christmas at Westminster Abbey and had the most amazing time. It is the midnight (of Christmas Eve, going into Christmas Day) service. We arrived around 10pm to queue up, you enter the church around 11, the service starts at 12. My husband and I ended up getting moved from our already nice seats to the wooden Choir benches in the center of the church - sitting amongst Lords and Ladies, just down from the Westminster Boys Choir (one of the best boys choirs in the world). It was magical. Hyde Park has a Christmas fair/market. It was nice to explore, there's an ice-skating rink. The lights on Oxford Street are pretty, Harrod's is lovely, the Cartier shop (on Bond, off of Piccadilly) had Christmas trees in all the windows. The decor is nice all around. We were somewhat limited on the sites due to the holidays, and we walked a lot - but we love walking and it was a mild Christmas anyway. Since you have a few days before and after you should have plenty of time to visit the places that will be closed on Christmas and Boxing Day. Have a lovely Christmas!
Great Christmas displays and cheerful holiday shopping along Oxford and Regent's Street, in Covent Garden and along the King's Road - and actually anywhere there's excellent shopping all year. Some of the displays really dazzle! Last year we loved the immense gold record and album covers decorations along Carnaby Street - too cool! Or take in a pantomime! Wonderful crazy amazing English Christmas tradition. There will be a Winter Wonderland to explore in Hyde Park. Kew Gardens is planning something BIG for Christmas 2013. Search out a concert at a church and listen to some carols. Hang out in a pub that has festive decorations and have a hot buttered run - very Christmasy! Hampton Court decorates in sumptuous Tudor style for the holidays, and there are special events - they have an excellent website. There is TONS to choose from. After all, England is where our modern conception of Christmas was really developed - and going to London is like going to The Belly of the Beast for cozy, sparkly, festive, CHRISTMAS-Y goodness and cheer.