I really do think that the best place for a photo shoot with family and children is out of doors preferably in a park where there are no dogs around to scare them. They really do get such a lot our of being outdoors and having no boundaries that they will relax and become themselves really quickly. The light in London is actually really good as it's mainly overcast...yes it really is better than being bright sunshine. You get far better softer lighting when it's cloudy and far better images out of it,so don't worry about the weather as I only ever have to rebook a shoot once in a blue moon.
To book a family photographer London has a lot to offer but I can assure you I have a lot more experience than most and will take your images to the next level