His birthplace according to his prison records was Aspley Guise which is in Bedfordshire and all his criminal records, he was convicted 4 times totalling 10 years and 7 days, are from the Bedford Gaol Register. Sadly the 1851 Census provides little help as none of the children's births, with the possible exception of Mary Ann, were registered as far as I am aware.
BAC3
A couple of thoughts here.
If you walk along Weathercock Lane the houses on one side of the road are in Aspley Guise and in Bedfordshire and on the other side are in Woburn Sands (now part of Milton Keynes) - Aspley Guise and Woburn Sands (and Aspley Heath where St Michael's church is located) are all contiguous.
Go back to the 1840s and Woburn Sands doesn't exist, the housing that was in the vicinity where the new railway station called Woburn Sands was built in were part of the parish of Wavendon, in Bucks. Wavendon was a large parish and had a number of "ends" and this area was known as Hogstye End but later became known as Woburn Sands.
St Michael's was built in the 1860s and prior to that time the nearest churches for baptisms if you were born in the area would be St Botolph in Aspley Guise or St James Husborne Crawley (both Beds) or St Mary's at Wavendon or All Saints at Bow Brickhill (both Bucks at that time).
The area was a hotbed of non conformity and there was a Friends meeting house in what is now Woburn Sands.
So Maybe look at Bucks as as well as Beds and also non-conformist records for him.
The other thought is the possible surname Buck - this makes me wonder if he had Traveller roots, there is a Gypsy Lane in Aspley Guise and an area up in Brickhill Woods, an easy walk from Aspley Guise and Woburn Sands, that appears to have been a permanent ("at all times occupied by Gypsies") Traveller encampment.