Every season is rainy season in London!
Actually that isn't true, but in November and December it does stay rather gray and dark. The hours of 'daylight' are rather short...sunrise at 7.30 or 8, sun's gone again by 4. The temps here in London are fairly moderate, so it won't feel too cold....usually doesn't get down below 40 or so. But it does get really, really windy! So it sometimes feels very cold. Add a bit of the famous London 'rain' (which is more like walking through a fine mist machine...umbrellas are useless!) and it can feel pretty nasty.
What you do need to bring is lots of layered clothing. The heating here in the UK is different to that in the USA and it does take some getting used to! Bring sweaters, t-shirts, hooded raincoats, good waterproof walking shoes, etc. But there won't really be a need for a big thick winter coat.
Hope that helps and hope you have a great trip!
