The obvious place to go for Christmas markets is London, because there is such a wide choice. You can see a full list of London markets here - the Southbank market is one of the most popular because it creates a lovely German-style atmosphere, plus all the Southbank attractions are nearby (as well as the Real Food Market and a chocolate fair!) But if you have children, they would enjoy the Hyde Park Wonderland more (you could visit both as they are not far apart).
Outside London, famous markets include the Bath Christmas Market - very popular indeed, but gets extremely crowded so if you want accommodation you should book now. If you want a touch of British tradtion you could try Gloucester Quays, which has a strong Dickensian/Victorian theme. Portsmouth has a similar style. Wales also has lovely markets - Caerphilly is one of the most popular.
You can find a list of the best Christmas markets around the UK here.