I live in London but often go back to Paris and travel to Milan in the winter. I really don't expect snow more than one week or so in December / beginning of January so I wouldn't worry about that. Rain boots and Uggs are a must for me though (not the most fashionnable shoes for sure but in London I can't really live without them as it gets really cold) Other than that I'd say a good pair of leather boots, scarves are a must as well. My favourite coat for the very cold weather is sheerling one from All Saints (but a little pricey as I got it on sale). And then black tights if you're wearing dresses (I really like the 100 den from H&M) and warm jumpers (I like some of the Topshop ones. In general, the British high street has a lot of nice jumpers, have a look on ASOS maybe?)