Hope I'm not confusing matters. Like you I wondered about Joseph's parents being William and Mary nee Sloan, but if I have the correct 1871 Census, I could not see how Joseph could fit into the family ie.
Possible 1871 Census @ Frizington, Cumbria .. Twigg .. John, head, 63, bo Staffs; Mary 58, bo Ireland with 4 children - Joseph, Alice, George + William (aged 13 to 29) (the eldest an Iron Ore Miner) plus 3 Grandchildren – John, George & William (3-6 yrs) bo Ireland.
Possible 1881 Census @ Arlecdon, Whitehaven, Cumbria .. Twigg .. William, 37, Iron Miner; Mary S, 25 .. both bo Ireland with 3 children – John - 17. George - 15. William - 13 .. all born Cumberland + Elizabeth, 7; James, 6; Mary, 5; Thomas, 4; Sarah, 3; Margaret, 2; Martha 0 (ie 7 Children in 7 years).
Perhaps your other information on this family explains the gap in the 2 groupings above.
It also looks like the same William above in 1901 Census at 118, Frizington Road, Arlecdon, Whitehaven, Cumberland, England (Head of household being a Mary Gambles) where he is described as William Twigg, Visitor, Widower, Male, 59, bo 1842, Diamond miner with numerous other Twigg visitors.
Cumberland newspapers have a number of articles of the Twigg family of Twizington if they prove to be attached to your Joseph.