I’d recommend you also consider Herne Hill and Dulwich. Some of the best state schools in the country are there (the Charter North Dulwich and Jessops Primary make the Sunday times top schools in the UK consistently based on their results. Judith Kerr Bilingual Primary / Dulwich Village Infants & Dulwich Hamlet / Harris Primary ED are all outstanding with very strong results.
Top private schools and prep schools are also concentrated nearby- Alleyns, James Allen’s Girl’s school are both highly selective and admit girls. Oakfield, Rosemead and Herne Hill School are other local options for your daughter.
Dulwich Village is the ‘leafiest’ part of the area and is 15 mins direct train to London Bridge. Nearby East Dulwich is full of young families, pubs, indie restaurants, cinema shops etc. Herne Hill on the other side of Dulwich has transport links into Victoria / Kings Cross / Blackfriars and has nice family amenities like Brockwell Lido and the Herne Hill Velodrome.
The area benefits from several great parks and playgrounds - Dulwich Park which is Grade II listed and has a boating pond, tennis courts, playgrounds and basketball courts, Brockwell Park, Belair Park, Peckham Rye Park (which has a new fantastic kids playground, skate park and Adventure Playground for older children). Kids like walking and playing in Dulwich and Sydenham Woods and Goose Green playground is also an option.
Other family amenities in the area are the Horniman Natural History museum which also has a small aquarium, a butterfly house with real butterflies flying around, and a small farm with chickens, rabbits, goats, sheep, etc.
Dulwich Picture Gallery has great exhibits and lots of activities for children on the weekends.
Good luck with your search. Hampstead is lovely. Given you like Richmond, Blackheath rather than Greenwich may be more your taste though Greenwich is lovely.