I would echo what previous posters have said and have a look at SE23. It is very family friendly (all state schools are good and above), has a nice sense of community. The Horniman museum is a fantastic resource for children and has a lovely farmers market on the weekends. Connectivity to the City is great. I work in the city myself and have multiple options (Southern to London Bridge and walk across / overground + DLR / overground + Elizabeth line) all about 35 - 40 min door to door if I time it correctly. If you did want to get to Dulwich Park/ Dulwich Village/East Dulwich on the weekends, there is great bus connectivity from Forest Hill too (185/176/197 depending on where you want to get to).