This is what I think…
From the 1871 England and Wales Census, 20 Ann Street Gateshead, County (Co.) Durham.
Those born in Newcastle-upon-Tyne, Northumberland: William (head) age 42 Bolier Maker; Isabella (dau) 16; Jacob M (son) 14; Ester M (dau) 11 and David Wm. (son) 9.
Those born in Co. Durham: Jane Swan (wife) 37, b. Houghton-le-Spring; John W (son) 18 Boiler Smith (?), b.Felling; William A (son) 4; and James G (son) these 2 boys born in Gateshead.
Also, I think William married Jane Scorer (born c. 1833/4) on Jan-Mar 1850 in Durham.
Hope this helps…