Bennet was the second of three sons and fourth of eight children of Charles Bennet, 4th Earl of Tankerville, and his wife, Emma (1752–1836), daughter of banker Sir James Colebrooke, 1st Baronet.
On 9 March 1771, he married firstly Mary Colebrooke, daughter of Sir James Colebrooke, 1st Baronet and on 26 May 1783 secondly Martha Catherine Carter, daughter of George Richard Carter.
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The Bridgeman Baronetcy, of Ridley in the County of Chester, was created on 12 November 1773 for Orlando Bridgeman, Member of Parliament for Horsham and younger son of the 1st Baronet, of the Great Lever creation.
For over 50 years Copley has been the partner of John Hugh Chadwyck-Healey (born 1922), grandson of Charles Chadwyck-Healey, 1st Baronet.
Meanwhile his brother, James had bought control of one seat in another rotten borough, Gatton in Surrey, for £23,000, and was also sitting in Parliament.
Pollock was the fifth son of Mr George Pollock, fourth son of Sir Frederick Pollock (1st Baronet, of Hatton) (elder brother of Field Marshal Sir George Pollock, 1st Baronet, of The Khyber Pass).