London Weather ( BBC ) is probably best known for two things: rain and fog. Contrary to the popular historical image, fog is a rare occurance, sometimes seen early on winter mornings but usually dispersed by the time people get to work. Any fog that does appear is usually high in the air and not at ground level, so the only thing it will affect is your view from the London Eye. Rain can happen at any time of year and sometimes without warning, either being a short burst of heavy rain, or prolonged light rain lasting a whole day and sometimes longer. However despite all of that London is a remarkably dry city, and you will often see stories in the local media about water shortages in summer months. In fact it only gets about 25 inches of rain each year, a lot less than say New York or Rome, and only a little more than Jerusalem, considered to be a desert city.
Summer can vary from being cool to unpleasantly hot and sticky, though is usually a comfortable medium. Like the rain, this can change rapidly. Much of London is ill-prepared for very hot days, so buses and trains can be an unpleasant experience at times. Most shops and restaurants are air conditioned, though hotel rooms frequently are not.
Spring and fall (known as autumn in the U.K.) can be nice, but the winter months can be brutal. Visitors from the mid-West of America will find London winters refreshingly warm - snow is very rare - but the constant greyness and rain will lead to depression. Day after day of coldness, cloudiness, and darkness may sound daunting, but remember that there is plenty to do indoors in London between the huge variety of museums, shopping, shows, and restaurants.
Because nobody in Britain pretends the winters are "mild", restaurants, pubs and buildings you enter to enjoy spending time inside are well heated. Many Mediterranean destinations are cold in the winter but refuse to admit it! So London - where you can hop from gallery, to cafe, to music venue to restaurant, is a good place to spend time in the winter.
In the winter, make sure to stay in a truly central location....waiting for a bus in cold wind and driving rain, just to get to the interesting places is a waste of a traveller's time, although this said, a tube ride in the rain is perfectly dry and will be more pleasant than one in the heat of summer, where carriages are often packed and boiling. Make sure you carry a bottle of water as every summer Transport for London report of people collapsing in tubes from dehydration! It pays to think ahead, so bring an umbrella and a bottle of water!