I have received a few helpful pointers following my previous post, most of all from Giles Colchester, a descendent of the Nettletons who knows more about the school than I expected to find out.
Courtesy of him, here are four picture of what Francis Ommanney called that "huge, red brick, hideous house and the level expanse of playing field that lay behind it":
It turns out that Mrs Nettleton's school at Stonefleet was destined to become Feltonfleet, which moved to Cobham, Surrey in 1917, to escape the airship bombs. It is still there. I ought to have known, as my brother-in-law was a pupil. It's funny how these things turn out.
January 2019: a further update. For photographs of some of the Feltonfleet school sports teams from the 1908-1911 period, please see https://www.flickr.com/gp/sjm_1974/39eQu8
January 2019: a further update. For photographs of some of the Feltonfleet school sports teams from the 1908-1911 period, please see https://www.flickr.com/gp/sjm_1974/39eQu8