The 1901 census of New Mills Street, Walsall (RG13/2704 f105 p28), and the 1911 census of 104 New Mills Street, list him as the son of Frederick Stanier, presumably by his wife Agnes. But his birth certificate shows him to be the son of Mary Stanier, of 87 Bradwell Lane, Wolstanton. The birth certificate has the father's name and occupation left blank, implying he was illegitimate.
The 2nd given name Boult is unusual, and may imply parentage. This draws attention to Albert William Boult; the Boult and Stanier families were neighbours in 1891, both living in Bradwell Lane. The Boult family was still in Bradwell Lane in 1901, suggesting the families were still neighbours in 1896, when Bert Boult Stanier was conceived.
Albert William Boult died in the DEC qtr of 1897, aged 26. Mary Stanier, Frederick and Agnes's eldest child, was then 23. It is not a great stretch to suggest that Bert Boult Stanier was actually the son of Albert William Boult and Mary Stanier, brought up as the son of his grandparents.