Dulwich: (East / West and the Village)
The commute would be 10 min on the bus and it has a number of outstanding primary schools. Harris Primary and Dulwich Village Infants and Dulwich Hamlet are all rated outstanding by Ofsted and Heber is also extremely popular with strong academic results and great extracurricular activities. Heber is the most oversubscribed LA primary in Southwark.
Heber and Harris are heavily oversubscribed for their main entry points in reception but as families tend to move around, there are occasional vacancies in higher years. I know Heber has spaces as a number of families have relocated out of London. Once you have your older children there your youngest will more easily get a place as siblings get priority so you don't need to necessarily live super close to the schools.
The area, particularly East Dulwich, is extremely family friendly with lots of young families and family activities (ballet, kids tennis, swimming etc), good shopping, restaurants, a cinema, museums (Horniman and Dulwich Picture Gallery) and great parks / woods.
Herne Hill is further from the hospital but Judith Kerr is the best school in that area and offers German bilingual instruction. It is one of the top primary schools in the country (by academic results and value add scores) and the has really happy and polite children.
Feel free to PM if you want to know more