Bedford was educated at Westminster School, and proceeded to St. John's College, Cambridge, admitted as sizar (age 17) in May 1724.
[1] As a result of his nonjuring principles he did not take a degree as that required an oath of loyalty to George I.
He was admitted into priests orders in the nonjuring Church of England by Bishop Henry Gandy on 27 December 1731, and became chaplain in the family of Sir John Cotton, with whom he afterwards lived at Angers.
Here Bedford prepared an edition of Symeon of Durham's De Exordio atque Procursu Dunhelmensis Ecclesiæ libellus, from what he supposed to be an original or contemporary manuscript in the cathedral library; from the same manuscript he added "a continuation to the year 1164, and an account of the hard usage Bishop William received from Rufus", and he prefaced the work with a dissertation by Thomas Rudd.
His will left sums to relatives and other nonjurors, including £20 to be distributed amongst the children of Dr Thomas Deacon, a nonjuring Bishop in Manchester.