I suggest having a hard think about how much house you want, and how important that is relative to niceness of area. It's a personal choice, hopefully with your budget you can get something you find acceptable on both counts (within an hour commute).
Personally I valued having a good commute, although when looking an hour door to door was acceptable, I did have a strong preference for under 45 minutes. I think if you are used to being on the tube, and get irritated waiting 3 minutes for a train, south of the river transport can seem limiting. In terms of commute, east London would be the best bet. Good commute means more time with the family, so it's not all about personal comfort. However, if you do not like the Essex border vibe of East London (particularly Wanstead and South Woodford) then that may be more important than a good commute.
I suggest spending some weekends checking out areas, shops, cafes, restaurants , green spaces etc. Also consider where the properties you can buy are relative to these amenities and transport and of course school catchment.
When I looked, on a lower budget, I did not like the Nightingale estate part of wanstead/south woodford. I found it endless streets of suburbia, not actually that near amenities and with road noise from the North Circular. I suspect there are areas like that in all the locations suggested, so don't rule out an entire area based on not liking the vibe of a single sub area. i.e. there are other areas of South Woodford and Wanstead which I would have happily considered.
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Good luck!