Hancock was born in Staffordshire, studied under Simon François Ravenet, and was at first engaged as an engraver at the Battersea Enamel Works under Stephen Theodore Janssen.
He was one of the proprietors of the works from 3 March 1772 till 31 October 1774, when he sold his share, after disputes with the other partners.
He retained, however, till January 1804 his property in a house built by Richard Holdship on the works, which he had purchased from the mortgagees in 1769.
Hancock's work often involved garden-scenes, milkmaid-scenes, and figures and half-lengths (especially of Frederick the Great).
About 1796, in Bristol, he drew small crayon portraits engraved by Richard Woodman for Joseph Cottle's Reminiscences: of Charles Lamb, William Wordsworth, Robert Southey, and Samuel Taylor Coleridge.