You absolutely do not have to be moving in an area for years before you apply! Bleuggh meant look at the distance from which schools took children during the last few years, to get an idea as to how close you need to live.
Applications for primary school are submitted in the January at the beginning of the year they start school.
London is STUFFED with good primaries.
I would start with an area that you would like to live in, it has the right transport, links to friends etc.
Then look at schools in that general area, and their admissions criteria.
Every borough publishes a booklet (online) with all the schools and details. Check the admissions criteria.
It is also worth looking at what secondaries are available in the area.
Loads of places with good primaries and secondaries. Dulwich, (Dulwich Village schools, Rosendale, and then Charter, for example), Catford : Kilmorie, then Sydenham Girls or Haberdasher Aske, ( has a new name now, I think), Streatham: loads of great primaries then Dunraven, Balham: Telferscott...
You don't have to live in the borough your child goes to school in, if you live close to a borough border your nearest school might be in another borough. That doesn't matter: only the admission criteria count.