Hi I’m sorry to say you probably don’t have much chance of getting into Broomwood or (I believe) Oliver House. We registered my son for Broomwood when he was a couple of months old, and he has not been offered a place for September 2020 - they have a ballot system and I believe around one in three got a place when they did the main offer last year (don’t quote me on the exact figures!). We have actually taken ourselves off the waiting list and I’m sure other parents will have done the same after accepting places elsewhere - but I imagine it will still be quite a long waiting list. last time I checked they were also planning to merge in children from a second site so will be particularly limited with places this year.
I also know a family with a child at oliver house - and the sibling will not be going there in September because she didn’t get a place.
You may get lucky, particularly if you’re persistent / keen, but I’d definitely consider other options.
Local fee paying schools that (i believe) are easier to get into are Dolphin school on Northcote Rd and eveline day school near Balham.
Near you there’s also bonneville primary school, I don’t have experience with it personally but I’ve heard good things. Otherwise Ravenstone school just the other side of Balham - this is probably where my son will go, I like the feel of it, it’s not as close for you but has a large catchment so you’d maybe get in there. Happy to post more details if useful. I’ve spent a lot of time researching local schools!