Hi. The proof or not would be in Athol's birth cert. - he was only 8 wks old at the time of the 1881 census. As for Stuart, don't think too badly of him. If you were born into the lower classes and wanted to better yourself that usually meant trying to move into a higher level, and that was very difficult - class prejudice in the British Empire was very strong in the 1900s. There was also a social stigma attached to illegimate children. Quite a number of people dealt with it by hanging their names, claiming they were born into a higher social level, or by emigrating and reinventing themselves. I have a number of relatives who did that - one was born in 1815 then disappeared entriely until the 1881 census when she popped up with a completely different surname claiming she was a widow - which she wasn't. It makes it difficult for us researchers, though.
You mentioned William Hugh Berners. You have probably flushed him out by now but he was, as I mentioned, born in 1846 the son of William Berners (1797-1855). He married Edith K Townend of Hove, Sussex in 1866. There are, as you know, no other marriages of a William Berners in the period to 1910.
1871 census - William & Edith are at Shrapnel Park, Byfleet, Surrey (probably Schrapnel Park) and he is a brewer.
1881 census - Edith appears staying with her mother, Elizabeth Townend, in Hove, with her daughter. It looks as if she is living off rents from property. Like you, I'm unable to locate WH.
WH Berners turns up in the 1891 census in East Portlemouth, Devonshire (name recorded as Burner).
1901 census - living on own means, Corsham, Wiltshire with wife.
1911 census - in Southwick, Surrey with his wife, private means, one child still living.
He died in 1929 aged 83.