Hello,
We had the same concerns a few years back when we moved to London and we decided to go to Wimbledon a bit further from Clapham, but it ticked all the boxes for us. What it is was really important was the support and advice we got from a company called SecurelyBe...in London, the lady that helped us through all the settling was really good and thoughtfully with our needs of info and all the million questions we asked them even before our arrival, I felt heard as she relateed to my situation.
Have you considered to get this kind of help? I have been in a few relocations myself in the past and when you have children you need help to deal with the uncertainties in lots of different areas.
Here is their website just in case could be your "cup of tea" ?
www.securelybe.com
Good luck with the move, London is a great place to be if you know how things work and what to do