Tube services, buses and trains will completely cease operation on the Christmas Day. There is no hope of getting around the city of London using public transport on Christmas. Public transport stops operation or offers very limited services starting from 24th evening (Christmas Eve) up to the 27th.
Taxis and minicabs are the only source of transport in the capital city. If you are travelling from the airport to Central London you may book your airport pick up in advance through online portals. Taxis will be in very high demand on Christmas day. If travelling on Christmas is inevitable then it is better you plan your journey well in advance. Minicab