Unlike the US of A where many places remain open on Christmas day London likes to take a deep breath and be with family for the holiday. It's rather refreshing. So on the 23rd and 24th (Xmas Eve) jam in as much as you can then relax and reflect on the 25th.
Head to Traflagar Square to see the Christmas Tree, walk down Regent and Oxford Streets to see all the store windows, the lights and the madness of Xmas shopping. Best time to take a London Bus after sunset. Sit upstairs with the nearly floor to ceiling windows as they move slowly very slowly along Piccadilly or Regent Streets. The lights are gorgeous.
On the 24th be first in line to see the Tower of London. Couple of hours. Photo ops at Tower Bridge.
From there walk up to the glass covered Leaden Hall market which will be festive. Have some coffee and a nosh. Look at the architectural difference between the Market at the adjacent Lloyd's of London building. Then stroll up Bishopgate to Spitafields Market for a look see. From there up and over to Brick Lane. Both will be jammed with last minute shoppers but worth the time and effort IMHO.
From there take the tube (Algate or Algate East) or walk (it will be a jaunt but will truly open your eyes to all that is London) back toward the Thames.
Couple of Pubs to pop into for lunch on the 24th.
If from Spitalfield's you were to walk to the Aldgate East tube station and kept walking on CommonWealth (believe it changes name after crossing WhiteChapel) for about 8 minutes to Prescott Street and turn right. There's a large multi story brick building (WPA looking) that consumes most of the block but just past it, mid way down the street you'll find the Princess of Prussia pub.
Says Truman Stout above the door and something about Burton Ales. Anyway, as of November of last year it was a traditional old school pub. Not a wine bar or gastro pub. Walk in and head past the bar on your left to the room into the back where the tiny Christmas Tree is located. There are chairs and tables. Great spot to rest weary feet and enjoy a pint or two.
My other favorite pub is down the road from St Paul's. It's called the Blackfriar's and is across the road from the Balckfriar's Underground/Train Station. Beautiful Art Deco pub!!!
That night maybe take in the traditional silly Christmas panto at the Wilton Music Hall if not sold out.
If up for it on the night of your arrival, the 23rd, maybe a theatre performance. After you've arrived and settled you could tube over to Leicester Square to the Half Price ticket booth. You can now check online what tickets might be available that night. http://www.tkts.co.uk/leicester-square/
If per chance your accommodations are anywhere near Hyde Park and you are an early riser on Christmas morning the Peter Pan Cup race is held at The Serpentine. Think lots of pink bodies plunging into the freezing cold water, quick swim and some warm alcoholic beverages. Then stroll around the park as Londoners emerge from their Christmas gift giving and enjoy the park, especially if the weather includes some sunshine.
Christmas will be quiet. If weather is good walk, stroll, meander or ride a bike. In fact a great way to see the city is to ride a bike as traffic will not be insane. Look for the Barclays or Santander bike racks that are all around London. Cheap and easy to do.
Some pubs will be serving Xmas dinner and many non christian cafes will be open for business as usual.
I love London at Christmas. Enjoy!