The William Douthwaite (wireman b abt 1874) is possibly on the 1881 census as William P Douthwaite, the 7 year old son of Hannah Douthwaite (widow). She is a lodger with the Nuttall family in Liverpool. He has a sister Mary E Douthwaite born abt 1865.
Now a curious thing, searching for Mary and mother Hannah on 1871 census, brings up a Mary Elizabeth with mother Hannah. Mary correct age born Liverpool, Hannah correct age born in Melmerby, Cumberland on both censuses. But Hannah is already a widow in 1871, so who is William's father? Mr Douthwaite, if he existed, had already died by 1871 if the census entry is correct.
FindMyPast has marriage record of this William, confirms the same one as the service record (occ wireman married in 1902). He says his father was George Douthwaite (deceased) father's occupation: Policeman. Marriage took place in Birkenhead. There is a death of an Ellen M Dowthwaite in Birkenhead district in 1927 age 54 - this age matches her age (30) at marriage.
A 90 year old William Douthwaite died in Birkenhead district in 1962. May be this William, age is close, agian depends on the informant's knowledge.