Most of my trips to Europe have been in the colder months of the year. For the most part, in the places you mentioned (London, Paris and Italy), the temperatures are rather mild compared to, for example, the Northeast US and Canada. Actually, you may find the weather similar to the Seattle area at this time of year. Snowfall doesn't usually accumulate enough to cause transportation issues. Of course, if you go to the French or Italian Alps, different story altogether. In the places you proposed, I would expect to see more rain and fog than snow. The weather tends to get harsher farther to the east, however.
As for popular tourist attractions, most remain open all year. In winter, they often operate at reduced hours, but because crowds are usually much lighter, you can see much more in less time. With the exception of the holidays, expect few lines.
Also in the winter, the indoor cultural seasons (symphony and opera) are at their height, whereas most go on hiatus during the summer.
Finally, if you are interested, the Alps are a winter sports paradise.
The only real downside to winter travel to Europe, in my opinion, is that with the vastly shortened hours of daylight, exploring the countryside becomes more difficult. The cities, of course, are well lit, so this isn't a problem.
So overall, I say go for it.