Osbaldeston was born at Hunmanby, Yorkshire on 6 January 1690/1,[1] the second son of Sir Richard Osbaldeston and his second wife Elizabeth, daughter of John Fountayne of Melton, Yorkshire.

Two of Bishop Osbaldeston's brothers (William and Fountayne) went on to serve as MP for Scarborough, their grandfather's former constituency.

He was educated at Beverley Grammar School, and matriculated at St John's College, Cambridge in 1707, graduating B.A.

He was not highly regarded as a bishop, but he was translated to London in 1762, "to nobody's joy that I know of" according to Richard Hurd, while Archbishop Thomas Secker considered him "in every way unequal to the situation".

As Bishop of London, he objected to a plan to commemorate a former lord mayor with a statue in St Paul's Cathedral, despite Archbishop Secker's approval, on the grounds that such monuments were not part of Christopher Wren's design.