In my opinion as your girls are in higher school years, I believe you will have a lot more choice of schools than those families who start primary from reception and secondary at Year 7. In those years top priority being distance from home with exception for church schools, criteria being religion/faith.
For example, two popular primary schools btc are near Clapham Junction but not close enough to be in the catchment area for an acceptance place from reception. However families in these schools do move out of the area and places become available in higher years, which you may be eligible depending on their waiting list. Having said that, Wandsworth have lots of great primary schools. I don't have any personal experience of John Burns so cannot advise.
Are you able to apply for a school place based on faith, attend church regular etc.
St. Cecilia's Secondary Southfields is a top choice https://www.stceciliaslondon.com/
and Holy Ghost Primary Balham http://www.holyghost.wandsworth.sch.uk/
. Not sure of their waiting list but if you prefer a church school, give them a try.
All the very best with your move and New Adventures here in Sunny South West London