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?