There is a lot of movement of people in this area, so don't panic. However you do need to think about your wants and needs.
Why are you moving to East Sheen? It's lovely, but it sounds like you need a state school. A lot of people in the area are frankly loaded, so pay for private schooling (for secondary.).
RPA is ok, but behaviour of the kids has been a problem at end of day. Especially at the bus stops on the south circular. I wouldn't move to the area to target that school, but it's not a bad school, just not excellent.
Would you pay private to have a place for your child?
Are y religious? Can affect school options .
If not the next most likely state school to have space is ARK in Putney (Wandsworth borough) It usually has space, it isn't very popular.
Christ's the other way in Richmond is currently full, but there is often movement as people move, if you are religious your chances are much higher to get a place.. Vary good for sports.
If Catholic Richard Reynolds school is close to Twickenham station so a lot of Catholic kids travel by train from South West London.
That year group was the biggest bulge year for primary intake so I can understand space at secondary this year is tough. Also didn't Chiswick school get a terrible Ofsted rating? So probably pushes more children into Richmond.
You could look at renting in Kingston, or Twickenham for some other options.