We had a similar dilemma when we had two young kids and lived in Battersea. I didn't want to worry about them if they walked to the corner shop, got shoes covered in dog mess and went to schools where drugs were a daily part of the routine. I had previously lived in Rickmansworth and my wife in Stanmore. "Ricky" is a bit grim now and Stanmore is mostly run by the Taliban.
So after much thought we decided to look South and targeted property within a 10 miles radius of Horsham, Sussex. After much searching we received details of a newish property in a smallish hamlet near a place called Loxwood. Detached in a wooded area, an absolute delight with a third of an acre. About the same value as our London place. Some cracking schools (state) in the area and there is even Christ's Hospital school in Horsham which I think is one of the wealthiest schools in the country.
BUT commuting to London was a pain. 10 min drive to Billingshurst and then the slow, 65 min, main line to Victoria. There was an option of driving to Godalming or Guildford and getting a faster main line to Waterloo. Driving to London was up to Guildford and then onto A3.
We now live about 6 miles further South but the M25 is close-ish and the coast is 30 mins away. Brighton is a fun place for our now grown up kids.
I never want to live in London again
dl