I have traced one of my lines back to William SAXBY b.1806 Elstow, Beds but can't go further. I believe he died 2Q1884 Bedford, Beds. He married Naomi BEEBY on 11 May 1827 in Elstow and they had 10 children.
I can find no other SAXBY family contemporary with William. In the censuses the name SAXBY is particularly prone to being mistranscribed in many variants: SARLEY, TAXLEY, SAGBY, LASBY etc. (As is BEEBY for that matter.)
The IGI has not proved helpful.
Does anyone have any early info on the SAXBY folk in Elstow or the City of Bedford itself?