From info on 1851 I guess that Jacob/Ann's g/s WILLIAM aged 5 was child of William/Margaret (tazz mentions a birth certificate, but does not name child ! )
So looking for marriage for William><Margaret in Northumberland ( or Co Durham- Bedlington was "within" Co Durham until 1844)
No sign of good candidate for marriage of William Mill(with varied endinds)
However if one " Cherchez la femme" ( I got French lesson at my school in Morpeth up through Newgate St lol) A Margaret Young marries in June Qtr of 1842 with Morpeth Reg District. ( Morpeth RD covered Bedlington as well as Morpeth- and much more)
Of the two grooms on same register page, one is a William MAIRN.
No sign of a William Mairn on subsequent censuses inNorthumberland !
I just wonder if the Mairn really could be Miller/ar

Michael