The son of Joshua Sacheverell, rector of St Peter's, Marlborough, he was adopted by his godfather, Edward Hearst, and his wife after Joshua's death in 1684. His maternal grandfather, Henry Smith, after whom he was possibly named, may be the same Henry Smith who is recorded as a signatory of Charles I's death warrant. His relations included what he labelled his "fanatic kindred"; his great-grandfather John was a rector, three of whose sons were Presbyterians. One of these sons, John (Sacheverell's grandfather), was ejected from his vicarage at the Restoration and died in prison after being convicted for preaching at a Dissenting meeting. He was more proud of distant relatives who were Midlands landed gentry that had supported the Royalist cause during the Civil War.