Ok so I know of people with children at the following schools:
Millennium - now outstanding (it wasn't always so), parents and children seem happy, mixed intakes but thought of as a good school, great facilities, regular homework. I hope it remains excellent.
Invicta - only downside is that it is close to a motorway but is outstanding and thought to be just an excellent school, great facilities, great head (or there was a great head) I always hear great feedback about this school. It seems like a lovely schcol.
Our lady of grace - difficult to get into but the parents I know who have children there are really happy and it seems to be a lovely school, I think you would need to go to church.
Sherrington - again really positive feedback
John ball - really positive again, great facilities, children and parents I know are happy.
Christchurch - do you mean in East Greenwich- not outstanding but good and parents I know with children there are happy
There is also a church school on the Greenwich peninsula and people who have children there seem very happy with it.
All saints - has a good reputation, very hard to get into. I know some parents there also who are really happy.
Halstow - Very hard to get into but I also know parents who are not happy with the school and think other parents are competitive and pushy which translates to children too, Now that could be just a poor experience from two separate families I know so it perhaps is not reflective of the school as a whole. I also know some parents were not keen on the academy process.
There are some good schools in West Greenwich too - James Wolfe is large but very well thought of, there is another church primary school in West Greenwich but I can't remember the name just now.
Meridian in Greenwich is also well rated and seems a lovely small school.
I am not up on the private schools as much but I have heard mixed reports on Pointers (check out the head) and Heath House due to lack of space. Blackheath Girls gets really positive feedback.
I think that in Blackheath and Greenwich the state schools are of a high enough quality that I would (if I were you) go state and not private. If you go state and the. Supplement any other activities on top but that is your choice.
Secondaries are more challenging but I know a fair few people with school age children and primary level education in Blackheath and Greenwich have improved and seem to be just getting better and better.
Greenwich and Blackheath are lovely places to live too.