This is a big part of SE London Mama224 (have you posted before asking about private/state schools in Chislehurst in the past few months? Are you coming to live in London from abroad and have one dd? I just seem to remember a similar request thread that several of us answered some months ago, but can't find it now...)
Anyway, in LB Greenwich, the top performing state schools at present are:
Christ Church on Shooters Hill - Church of England School
St Mary's, Eltham - Catholic School
Wyborne, New Eltham - non-religious state primary
Brooklands, Blackheath - non-religious state primary
You have to live close to these, especially CC and Brooklands, to be sure of getting a place, and you have to be a practising Christian at CC and a practising Catholic at St Mary's.
Can't really recommend anywhere specific in Chislehurst/Bromley but Chislehurst is a lovely place to live, Bromley in general fares well in league tables, so I'd say most of the state schools in that part of Bromley are likely to be good prospects.
But the way the UK state system works is you have to have an address near the school, and be able to prove it is your current residence, in order to apply.