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You can go onto the waiting list at any time, so if you can move right onto the doorstep for Dunraven you should go straight to the top of the waiting list and get a place reasonably soon. Perhaps! But it will be the same anywhere. Waiting lists are kept in order of eligibility, not in order of when you went on the list. The general catchment for Dunraven is up to .9 of a mile. This is the period when lots of people are on various waiting lists and places become available when someone accepts a school they had preference for.
Sought after state primaries are Julians and Streatham Wells - both have very tight catchments.
you'll probably get more answers if you post this in education or secondary education
the only secondary which is actually in Streatham is Dunraven, it seems to be well thought of but has a fairly small catchment for the higher ability bands afaik
You won't be near enough to get into Graveney on distance (it has an 'extension' stream)
If your dd has just turned 11 (i.e. in Y6) you've missed the secondary transfer process so will be offered whatever's left over, typically the schools no-one else wants I'm afraid and certainly not guaranteed to be your nearest. If both your older children are already secondary age, you'll have to see which schools have places - London is a fairly transitory place so people do come and go, but it's unlikely that both your children will be offered a place in the same school.
Private schools are just as much of a bunfight in South London, although again, it's possible that places may come up as people move away
Moving back to the UK and settling in Streatham. What are good schools for DD (aged 11) and DS (aged 12 and gifted) and DS2 (he will be just 3, but we should buy near a good primary school). Advice re schools, best areas etc all appreciated.