If he believes that he was definitely born in New Basford, with father John, and correct age, then that does limit the search area.
School records?
I was reading an article about New Basford, which lead to the workhouse, so looked at Hills there, as one does, 2 children were born in France, Francis and George age 6 and 4. Can see Francis in 1911 but not in-between (as yet). Just as an interesting fact, no connection implied.
We always have this problem of whether a child definitely knows where they were born, and why William did not specify Basford in the 1901 census
1921 census.
Sorry, these are just random thoughts, jotted down.
Did the professionals say - no, that's all we can do, or suggest other avenues?