We spent Christmas in York a few years ago, and then days before in Edinburgh. I love the Xmas fairs and your children will as well! We planned to stay put Xmas day and Boxing day. Christmas Eve was busy late with young people out and about...seemingly not with their families, which I thought was odd, but certainly provided much entertainment. My DH went to services at Minster Abbey. Xmas day was totally quiet. We had made reservations a few months before for Xmas dinner at a restaurant, which was fun. Otherwise, we strolled around and enjoyed the charming little old city. Afterwards, on Xmas we went to the cinema. Boxing day was busy with shoppers, etc. I think London activities would be similar.
Another Christmas we spend in Ealing(London) with family. Xmas Eve at pubs were much fun but your babies will be in bed by then. Xmas day was at home and out walking.
Clothes: All my winter travels to the UK, so far, were fine with an unlined leather jacket, leather gloves, scarf and earmuffs if really cold for them (note: I'm from New England). If it was particularly cold, I also wore a boiled wool sweater. We are outside most of the time walking or walking between activities. Rain rarely interferes or comes for an extended period, in our experience. If the kids are in strollers, which I suspect they often will be, just layer them up more. Pack up one or two of those stroller rain coats if you would feel better. Enjoy your kids and the traveling. It goes all so fast, and someday you will be able to have all adult vacations you can...even with them, but in the meantime, enjoy their childhoods AND travel. Wray