We've recently moved from Hampstead to Walthamstow. Quite a lot of my friends have also followed us out (from Hampstead, and also from SW London (Fulham/Chelsea).
Honestly, I feel safer here than walking out of either Belsize Park or Hampstead tube stations at night. There has been a few car break ins and a lot of bicycle theft recently - but as I understand it this is also the case in NW3/NW6 at the moment, and is not a usual pattern of crime.
There is a lot of support for independent businesses, the environment and cycling in our local communities. It is very much a community feel where we are (perhaps more so due to lockdown) and are all much more relaxed with the vast amounts of outdoor space we have access to.
I was told that recently that E17 has consistently been in the top ten places to live in London in recent years (I haven't actually looked into this myself, but it was from a friend who was moving here).
Public transport is good - Victoria line and Overground, plus a little further south or east is easy access to the Central Line. Westfield Stratford is a million miles better than Brent Cross!
I would perhaps (if you can) reconsider the four bed. We did and financially it's been a great move - we went for three bed, and are now in the process of extending - financially it's saved us around 150k (there's quite a price gap between 3 and 4 beds here and it was much cheaper to do the conversion ourself).
Our place was liveable in, but definitely not our taste so we are renovating and redecorating further - I do think that a lot of the properties here that are on the market are from older people so that's something you may want to factor into your finances too.
Regarding "pockets" - yes there are here, just like there are in Camden, but I certainly haven't found myself in them, or affected by them. There's bus route we take regularly and do sometimes walk which goes through an area that is similar to Parkhill Road NW3 area and I honestly have never felt threatened in the E17 equivalent. Did in Parkhill but that was largely due to the freaking moped muggers.
Oh, and prices are going up here as well (at least according to the new estimates we've had, and the number of estate agents we've had knocking on our door asking if we are prepared to sell)...
Hackney really doesn't fit your bill - it has less green space than NW3/NW6 - E17 has the green space that Hackney has on offer, plus a whole lot more.