Like you say Jo, on the 1851 census, the illegitimate children could be Major’s children
1861 Census: Red Hall, Lower Gornal, Sedgley. Major Hale, Head.UM, 27, Excavating? Labourer Mary Wakelam, live together.UM, 32, Nailer Esther Wakelam, Dau.Illigit, 5 John Wakelam, Son.Illigit, 3 Elizabeth Wakelam, Dau. Iligit, 1? (All born Sedgley)
But, if Mary married that Benjamin Bate in 1846, those illegitimate children would surely have had the Bate surname instead of Wakelam, as would Mary.
I think it would be best to archive this now, what are your thoughts?