Joseph was the son of Joseph Wilcocks Senior, Bishop of Gloucester, Bishop of Rochester & Dean of Westminster, by his wife, Jane, the daughter of John Milner, the British Consul in Lisbon. He was born in Dean's Yard at Westminster in Middlesex, on 4th January 1724. He was admitted to Westminster School upon its foundation in 1736 and was elected to Christ Church College, Oxford in 1740, matriculating on 10th June and graduating as a Bachelor of Arts in 1744 and a Master in 1747.

Possessed of a considerable estate, Joseph modestly devoted his property to acts of beneficence, and his time to study. He was elected a fellow of the Society of Antiquaries in 1705. While residing in Rome, his piety and benevolence won the admiration of Pope Clement XIII, who styled him the 'blessed heretic.' For the use of Westminster School, he prepared four books of 'Sacred Exercises'. He lived for some time in Barton, Northamptonshire, and afterwards at Ladye Place, next to the church in Hurley, Berkshire. He died unmarried at the Crown Inn in Slough, Buckinghamshire, on 23rd December 1791, and was buried in Westminster Abbey on 31st December, in his father's vault. He left behind, prepared for the press, a work founded on his residence in Rome, entitled 'Roman Conversations, or a Short Description of the Antiquities of Rome' (1794), which contains many autobiographical details. He bequeathed the second edition to Brown, his publisher. It appeared in 1797, with a memoir by Bickerstaffe, Brown's successor. Wilcocks was also the author of 'An Account of some Subterraneous Apartments, with Etruscan Inscriptions, discovered at Civita Turchino in Italy,' published in 'Philosophical Transactions' in 1763. Some verses by him appeared in 'Oarmina Quadragesimalia.'