There’s a funny song by Flanders and Swann about the UK weather. To some extent, it’s true.
The UK is a relatively small country. The north, especially Scotland has quite cold and snowy winters and pleasant summers. The west gets lots of rain, blown over from the Atlantic. The south west can be rather warm and sunny.
London is in the south east of the country. We do have some hot days in summer, maybe an occasional very hot day (which usually causes travel crises) but the days tend to be warm, dry and nice for walking. Summer can be hit with cold spells though.
Winters can be cold but are rarely bitter. London can go through a whole winter with no snow, and even in the coldest snowiest spells it is not bad (though it usually causes travel crises). It can be grey and dark, and many attractions can close for off season. If you go up tall buildings or rides, the view may be restricted.
Xmas does have street lighting and shop decorations, but they are often underwhelming. Theatres tend to close over the holiday, or have pantomimes, but the ones open usually have good availability.
The UK has a tradition for lots of rain, but it’s not that bad in London. There have been flooded roads on occasion, but this is unusual, as is thunder and fog. Sometimes there can be water rationing (oh, tap water is safe to drink).
Summer is a nice time to come, especially for boat trips and outdoor walking or park visits. Autumn is also good. If you are intending to have days out, eg to Brighton, summer is best. Coming out from a night at the theatre and it’s still daylight is a real treat.
On the whole, the extreme of weather are rare. The weather is ‘temperate’. Winter can be grey and a bit gloomy, summer very busy and expensive. Autumn or spring may be your best choice, especially as the kiddoes are still in school, not hanging around on the streets, and flights may be cheaper.