I've been in London during end Nov. All the places that I wanted to see were open (museums, Tower of London, side trips to Bath and Cambridge) but with shorter hours due to the shorter days. It would help if you looked up the opening times of what you're interested in specifically, for the rest of the major attractions you shouldn't find any closed doors. Xmas lights and bling in the Oxford St area was a treat! Weather was cold with a spot of wind - layers is the way to go.
I've been in Scotland during mid Mar. Although nothing was closed (shorter hours, yes), Edinburgh's whipping winds made me want to just tuck-in, and it took an effort to step out into the cold each day. Glad you don't mind the cold :)
Either place, I loved the fact that the tourist flow was dwindling down. Nothing beats that if you can put up with the weather. I've dreamed of summer in Europe but now I'm a shoulder season advocate :) Have fun!