Matches this Joseph in 1871, down as Bryers rather than Bryan :
Downs New Houses, Hetton le Hole,
William Bryers, 49, Coal Miner, b. Chorley, Lancashire,
Elizabeth Bryers, wife, 46, b. Evenwood,
Joseph Bryers, son, 22, Coal Miner, b. Toft Hill,
+ 6 other children aged 11 to 2.
Mary Lawns, Boarder, wid, 78, b. Cockfield.
Also this Joseph in 1851 :
West Cornforth Village,
George Lawns, son 21, Unm, Coal Miner, b.Evenwood
Elizabeth Lawns, dau 24 (28?), Unm, b. Evenwood.
Joseph Lawns, son, 2, b. Evenwood.
Down as sons and daughters, but are in a separate household to the family above them, so I assume their parent(s) were absent.
I suspect Joseph is the son of Elizabeth Lawns, who then later married William Bryers, and Mary Lawns is probably Elizabeths mother.
Only problem is William Bryers appears to have married an Elizabeth Rawlinson. Trimdon 4 Mar 1854. Both of Trimdon Colliery, Bachelor, Pitman, and Spinster. Fathers George Rawlinson, Blacksmith, and Thomas Bryers, Labourer.
Rawlinson - Lawns? Elizabeth Lawns from 1851 census does appear to have a father named George.
George Lawns married Mary Chaters, Teesdale 1818. Then 5 children to George and Mary : Ann 1819 Cockfield, Joseph 1822 Cockfield, Mary 1822 Cockfield, Betsy 1825 Evenwood, George 1830 Evenwood.
Would need some certs to confirm I think.
Alan.