All sorts of towns and villages to stay at in the Cotswolds. I like Cirencester - good public transportation to/from Heathrow, Oxford, Cheltenham. Roman remains (ampitheatre, roman walls) and a nice museum, pubs, A nice trip out to a nearby Roman country villa (Chedworth), pubs, a street market (check days), good shops, a lovely old Church, did I mention pubs?......a nice park along a river with good walking trails.
Broadway is a lovely village but quite expensive to stay there and you really need your own vehicle.
I can remember some cold, snowy Christmases in the Cotswolds on family visits - but it can also be very mild. Certainly it will not be too busy with tourists in December. Just bear in mind that away from London a lot of attractions may not be open between Christmas Eve and until after New Years Day.