Advice Winter in London and Paris: What to Pack?

I will be on the RS's London tour at Christmas and plan on taking a pair of leather boots with me. I'm also considering a short trip to Paris during the same period. I'm a bit unsure about what to pack for the winter weather in these cities. Should I expect a lot of snow, or is it mostly cold rain? Also, any tips on layering or must-have items for sightseeing in the cold would be super helpful. Has anyone traveled to London and Paris in winter and can share their packing essentials?