The artist Eppo Doeve painting the Martineau children at “Old Lodge in Terlow (Buckinghamshire),” Great Britain, July 1954, by Willem van de Poll, via Nationaal Archief (Netherlands).
I have not been able to find out anything about these siblings or the home. I cannot find an Old Lodge in Terlow, Buckinghamshire, or indeed a Terlow anywhere in Great Britain. The Martineau family is quite important in Birmingham (an ancestor of the Duchess of Cambridge was a Martineau), but apparently not in Buckinghamshire. Perhaps the photographer made some mistake in his notes.
The son on the right seems to have three golf balls between his fingers. A young amateur champion?
enclos*ure 
Interesting and rather odd that a British family living in southern England would choose an Indonesian/Dutch artist who, according to Wikipedia, primarily did cartoons. Perhaps the Martineau family had Dutch or Indonesian connections.
I wondered if a Dutch painter had mis-spelled the name of the town and checked for any variation that seemed likely. No luck, so still a mystery.
I really searched as well. I suspect there’s some small, but key, bit of misinformation in the Nationaal Archief labeling. Or, despite the ‘next generation’ quality of the painting, the family moved out of the house shortly after it was painted and never made it into local history. (Although, does ‘Terlow’ exist?)
Cindy, your source link should use this Nationaal Archief permanent URL:
http://proxy.handle.net/10648/aecb9a42-d0b4-102d-bcf8-003048976d84
Thanks! I put together this post some months ago, and I think the website changed the content of my original link.