The 24th, Christmas Eve, there will be few problems, everything runs as normal but with Saturday routes. After 19:00 there will be Sunday routes. Some tube-stations close at around 21:00. Airport Express-trains runs more or less as normal.
The 25th, Christmas Day, well this is not the day you plan for the big outings. There are no trains, Dockland Light Railway or underground at all this day. In the morning, everybody snuggle up under their Christmas tree opening presents, so even the London buses has taken a day off. There are some few companies running bus-routes but nothing really to rely on. Travelcards etc. will not be valid on these buses. This is the day for a nice walk in the park and London can offer several! If you do need to transport yourself, taxi will be an option but should be booked in advance.
Heathrow Express runs half hourly service between 06:30 – 14:00 after this a bus service will replace the train until ca. 21:40 and Gatwick Express has a bus-service every half hour from 06:30 to 16:30. There are no Stansted Express trains this day.
The 26th, Boxing Day, There are reduced Sunday routes on the tube and Dockland Light Railway with first underground around 08:30. Buses will operate also from 08:30 on all routes. There are only a few trains in Scotland and Southern England that operates as well as the Airport transfers like Heathrow Express, Gatwick Express etc. Stansted Express will be operating between Tottenham Hale and Stansted Airport so you have to catch the underground to Tottenham Hale which is in zone 3.
For more information, check out Transport for London’s Christmas and New Year information.
Have a nice vacation and a safe journey!