Hello!
Herne Hill and Tulse Hill have a dearth of local community primary schools. I would go for West Dulwich and live as close to Rosendale Primary as you can afford. It also has a nursery but I can't remember from what age, I suspect it has a very long waiting list.
Alternatively, I would look at Streatham Hill and see if you can live near to Hitherfield Primary School.
The reason I suggest Rosendale and Hitherfield is that they are both 2 form entry schools so it is a bit less of a gamble hoping to get into those, in my opinion, than some of the smaller ones. Rosendale is already "Outstanding" and I think Hitherfield is just "Good" but I think Hitherfield is on the up.
Herne Hill is a very nice area for new parents - there are baby groups, there is Brockwell Park, there is a lovely toy shop etc. The station goes to Blackfriars but not London Bridge. There is a nice private infant school there called Herne Hill School.
Tulse Hill is a handy station for commuting to either London Bridge or Blackfriars / Farringdon / Kings Cross. Tulse Hill itself is not as nice as Herne Hill, so houses are cheaper. But you can easily end up in a primary school black spot.
West Dulwich has lovely houses but I think they are expensive. There are some good private primary schools - Oakfield & Rosemead. Dulwich Park is gorgeous.