Apart from the chance of bad weather remember that it gets dark early so trips in the countryside are necessarily curtailed. My general rule of thumb is that winter is the time for visits to cities/towns.
Either stay in London and do day trips to places like Oxford, Cambridge, Canterbury or York. For York you would have to book ahead to get the best train fares.
http://www.nationalexpress.com/trainslanding.aspx
Or perhaps better, stay overnight in York. If the weather is good you could rent a car in York and visit some superb coast and/or countryside.
http://www.visityork.org/seeanddo/
Same goes for somewhere like Canterbury, nearer to London, plenty of countryside and old country houses/castles around there.
http://www.canterbury.co.uk/seeanddo.asp