Barbara,
This should be fairly easy, but as you already know, it isn't ! So....
From the start of civil registrations of BMDs in 1837, Wolviston (within the Parish of Wolviston) and Wynyard (within the Parish of Grindon) both lay within the Registration District of Stockton.
If William Kelly had been born in Wolviston (1891 info) or Wynyard (1881 info)
his birth would/should have been registered with Stockton ( in that era, Stockton's sub-districts were Stockton itself, Hartlepool, Sedgefield and Yarm)
I can see no sign of birth reg for a William Kelly ( even if his birth been c1838--- 1881 census has him aged 43 ! )
So was he born/registered elsewhere or in a name other than Kelly ?
Was Crook-born James there first child. It is possible they had an earlier child, born elsewhere ?
Have you spotted Mary Ann on a census before 1881 ?
Michael
Newcastle