You've probably seen it already, but there's a book written by John Jochimsen (available online).
He attended Cumberland House School circa 1935, which he describes as a "small private establishment in Grand Drive". He also describes the classroom: "had once been the drawing room of the large house" and how lunch was in another building, at the bottom of formerly a long garden.
All in all, it doesn't sound like a "formal" school building in the traditional sense. Hence the lack of online information, except for two 1935 newspaper mentions (Headmaster: JS Wardale). Probably the newspaper mentions which Shaun found.
UPDATE: Address of the school could be 42 Grand Drive, West Wimbledon, SW20. This is the address on John Stephenson Wardale's WW1 Medal Index Card, dated 1922. (Raynes Park is also known as West Wimbledon).