Hi janet75, I live in Blackheath/Greenwich.
You want to live near your school (especially if it is state in order to fall within catchment) but the bottomline is affordability re: housing.
The good primary state schools in Blackheath (Lewisham council) are as described by MrsJohnCusack. Bear in mind All Saints is Church of England so helps to know the vicar plus regular attendance of course .
The good primary school in Greenwich (Greenwich council) is Halstow. Bear in mind the catchment for all of them are tiny and you really need to live as close as you can to them to jump the waiting lists. I assume you will go on a wait list because you are joining midyear and the good schools are 110% most certainly oversubscribed. The priority on the wait list is (I think) by proximity so you leapfrog anyone on the wait list if you live, say, next door to the school gates. I would suggest renting to do this. As this is London, people always come and go and so there is a chance a vacancy will arise. If you really really want a place, the school advised me to keep close to them ie call them every now and then, speak to the headmistress ...
Check the Ofsted reports for their scores and write up (although Ofsted is not everything):
[http://www.ofsted.gov.uk/ here]]. For private schools, the Good Schools Guide book.
I imagine you would probably put your 4 year old in a less popular private school that has vacancies and on the wait list for a good state school or a 'better' private school. But this gets tricky in terms of where you where to live (assuming you want to be close the school) so renting is not a bad idea for a start and also in the current soft property market.
State secondary schools in Greenwich are generally dire bar one (I am not sure about Blackheath). So unless your dcs can get into the academically competitive grammar schools in nearby Bexley, Bromley and Dartford, which they will have to commute to, you are pretty much looking at private education for secondary.
Between Greenwich and Blackheath, as a broad generalisation, Greenwich is better for transport links to Canary Wharf, because it has the DLR. But Blackheath is generally closer to the good schools, both state (in terms of choice) and private. So you can decide which you prefer. Both would require a bus ride/drive to the Greenwich tube station (1 stop from CW station) so you always have that option from either location but will add at least 20 mins to the commute due to unpredictable traffic. Blackheath only has the mainline train (but a good line) - hence relatively isolated and more villagey. Greenwich has the mainline train and DLR, which brings the tourists, so more edgy and trendy.
I would prefer commuting to Canary Wharf from the south (ie from Greenwich/Blackheath) than from the North (ie from Maida Vale etc). The Jubilee line is jam packed from the north.