I'm headed to London in December. We land on the 24th. What can we do on Christmas Day and what will be open? I know public transit will not be running. Ultimately booked a hotel since responses to flat/apartment rentals were unsuccessful. We are staying in Southwest London near Battersea very close to the River Thames.
I can tell you how we spent last Christmas Day in London.
We walked through St, James Park about mid-morning with lots of Londoners. As it started to drizzle, we walked to Trafalgar Square (we were staying near Westminster Abbey) to take a walk at 11:00 with London Walks. There were 100 people there with umbrellas waiting for the walk to start. Yikes! So we wandered off back to our apartment. We had a lunch reservation at 14:00 in Soho. The rain had stopped so we walked there, then all the way across town to Marble Arch where we saw the new Star Wars movie. The house was packed so we were happy we had bought tickets online a week in advance!
So, get an apartment; make a lunch reservation as restaurants book up for Christmas lunch; take walks; see a movie. Of course there are church services if you like.... Also reserve for Christmas Eve dinner. Very busy in the restaurants!
if there is a Boris Bikes (Santander Cycles) docking station nearby that could be another option of transport that is available every day. Take great care though because you will mostly - except in Battersea Park - be sharing the road with bigger vehicles.
I've spent 2 Christmases in London. I live on the west coast of the states where sunshine reigns, stores stay open 24/7 and traffic never seems to cease.
Discovering that one of worlds greatest cities shuts down so that everyone can truly respect Christmas Day and be with family gave me great joy.
It a chaotic world its refreshing to experience such a "time out."
So in answer to your question, if weather is good be prepared to explore the neighborhood around you on foot.
If you arrive early enough on Christmas Eve and are staying at a hotel reserve for Christmas dinner if the hotel is serving.
If you are staying in an apartment check to see if any nearby local eateries are serving on Christmas and reserve a table. Some ethnic restaurants stay open or find the pub closest to your accommodation and find out if they are serving. I'd also stock up on food if you are staying in an apartment.
See if a nearby church is having Midnight Mass. Years ago, one of my more favorite moments in London involved midnight mass at St. Paul's. When we left the cathedral, a light dusting of snow had fallen, the bells were peeling, the glow of the street lamps....truly a "Wow, its Christmas and I'm in London" feeling.
On Christmas Day I'd stroll through Battersea Park. Families will be out, kids on new bikes, people walking their dogs, babies in new prams. Its a lovely way to see "real" London.
If weather is bad you can always watch Queen Elizabeth's Christmas Day message on the television.
Relax, enjoy the quiet, and meander.