Can't comment on SE London but we lived in Hackney for 5 years, on Victoria Park in the 'posh' bit. We moved to Suffolk 2 years ago with our (then) 3 and 2 yo.
We lived in the catchment for a wonderful outstanding primary school, and being around V.P with very little children was brilliant as there is so much to do and easy access to central London etc.
Then i looked at the secondary schools and they are, almost without exception, very poor/average in Hackney.
In the 5 years we lived in Hackney we were broken in to (police astounded, we had THE most secure house they'd ever seen but someone just shimmied up the front of the house and crawled through a small window 4 floors up with a 40 foot drop the other side), I was mugged for my handbag at 9am, an old lady was mugged in daylight outside our front door, taxi driver stabbed to death 20 yards from the front door, 14 yo girl gang raped about 50 yards from the front door, car broken in to twice outside the house. And that's the stuff that was very close to us, won't bore you with all the other.
Enough was enough. I just didn't want my children growing up somewhere where I seriously feared for my own safety never mind theirs. And I wanted space for them.
They will probably hate me as teenagers but there we are. When we moved here I read the crime report in the parish magazine, it said, "PC Plod reported one crime for the last quarter. Some dog biscuits and a dog collar were stolen from an unlocked car. PC Plod reminded all parishioners to lock their vehicles at all time.". Oh...I laughed...
London is a brilliant brilliant place. Just wasn't for us long term, in the end.