London is a year-round tourist centre, with few of its attractions closing or significantly reducing their opening hours in winter. Your best chance of good weather is, of course, at the height of summer in July and August, but there's certainly no guarantee of sun even in those months - plus it's when you can expect the biggest crowds and highest prices.
April/May and September/October are good times to visit London: there's a better than average chance of good weather and the queues for popular attractions are hours shorter. During these periods the cost of getting to London is cheaper and you might even find some decent accommodation. If you don't mind braving the cold and shorter daylight hours, the cheapest fares and hotel prices are available from November to March.