The 1871 census has William snr vet surgeon age 31 b Skirlaugh, Yorks, wife Jane also 31 b Arram, Yorks, son Henry Lonsborough Douthwaite 3, William 2 and Jane Elizabeth 1, all children born Beverley. Lonsborough as middle name could give a clue as to Jane's maiden name.
1881 has 6 more children, Henry not at home. Youngest son (7th) called Septimus aged 10 months. There is a younger sister Annie which would match the Annie in 1901 in Bournemouth.
By 1891 Jane is a widow (living on own means) , William and 3 younger siblings living with mother. William is a banker's clerk.
But there is a William dying in Beverley in 1923 age 54. Probate to Henry Lonsborough Douthwaite dispensing chemist who is definitely his brother. So it looks like this one is a dead end.