More speculation!
In 1841 Ann Coulson, aged 1, living in household of Ann Coulson( age 55), Newrick (age 20) and Ellen(age 20) -adult ages rounded down to nearest 5 years. Living South Hetton, The long Rows, Easington ( approx 3 miles from Pittington, birth place/child hood home of Ann Coulson/White). Relationships not given but seems possible Newrick and Ellen brother and sister. Ann daughter of Ellen?
From Family Search:- Ellen Coulson/ James Smith married 28 Dec 1844, Kelloe, Durham.
From Family Search:- 1851 James and Ellen Smith living Durham with daughter, Ann, aged 11, born Pittington.
From Free reg:- Newrick Coulson baptised 4 Feb 1816. Son of John Coulson and Ann Mowbray.
I accept there are a number of gaps but I do think it provides a reasonably cohesive narrative linking John and Ann( Mowbray ) Coulson to Joseph Herrinngton Coulson, but as I said previously, subject to closer scrutiny. Could be one long string of coincidences!