Well visiting London is a must, you could easily spend your whole visit there and never get bored! I don't live in the city, but close by. Gotta visit China town for the atmosphere and the scrumptous food. Don't stick to the tourists routes, follow the locals, they know where's best! Well perhaps go to the Hard Rock Cafe. Yes it is very cheesey, but one of my favourite places. They have a leather Bono jacket! Have a few drinks, then go across the road and see my pal Jimmy (who named me a dudette) who runs the vault. He's often very drunk around the Xmas period! You can sit on Freddie Mercurys chair, hold B.B Kings trumpet (i think!?!). John Lennons jacket etc etc.
Use the tube as least as you can, it may seem easy getting from place to place but walking you get to see more of London and soak in the views!
Use Easyjet Airlines for travelling around Europe, it's the cheapest around. If you book far enough in advance flights can cost little as 99p!
England is famous for its dreary weather! December isn't the coldest month but probably alot chillier than your used to! Around 10-15C average as a wild guess?