I would either go for Havering - the Easternmost London borough - or Leigh on Sea/Chalkwell (Essex coast).
The better areas of Havering are Upminster, Gidea Park and parts of Hornchurch. All have fairly decent high streets (albeit Gidea Park's being just a short stretch of Main Road), lovely green spaces (Raphael Park, Upminster Park, Langtons, Hornchurch Country Park, Thames Chase) and loads of kids activities and family events.
Havering is in London travel zone 6, meaning public transport, both locally and into central London is frequent, reliable and reasonably priced. Upminster is on the C2C line, going into London Fenchurch St, and out to Leigh on Sea and Southend via Tilbury. It's also the first/last station on the District Line. Gidea Park is on the new Elizabeth line, which is really fast into and beyond central London. Hornchurch lacks a mainline station but is on the District Line, there's the tiny Emerson Park station with a "push and pull" train to Romford and Upminster. There are numerous bus routes too.
I don't think you can go too far wrong with Havering primary schools. For secondaries, Coopers (religious) and Hall Mead in Upminster are highly sought after, and In Gidea Park Royal Liberty (boys) and Frances Bardsley (girls) are really well regarded too. A lot of children commute to the Essex grammars as well.
If you'd consider living further from London, Leigh on Sea has become very trendy in recent years, artisan everything, loads of cultural events, opportunities for water sports, the lovely Belfairs woods. Decent schools including grammars. Lovely Victorian to 1920's housing stock. It's on the C2C line so direct trains to Tilbury and London. This is definitely where I'd go if I moved away from Havering.