Speke was the only son of John Speke of White Lackington and Dillington, MP, a wealthy and influential Somerset landowner and his second wife Elizabeth Pelham, daughter of Robert Pelham of Compton Valence, Dorset. He married Alicia Brooking, daughter of Nicholas Brooking.[1]
Speke married twice after the death of his wife Alicia. He married Jane, the widowed daughter of William Hockmore of Combe-in-Teignhead, Devon in February 1732. Her former husbands were Palmer of Sharpham Park, Somerset, and William Pitt of Cricket Malherbie, Somerset. He then married, in July 1737, Anne Lady Drake, widow of Sir William Drake, 6th Baronet, of Ashe, Devon and daughter of William Peere Williams of Grey Friars, Chichester Sussex.[1]
Speke died on 2 January 1753, leaving two daughters, but was predeceased by a son and another daughter. By his first wife Alicia he had one surviving daughter Mary who was left £10,000, and by his third wife Anne he had one surviving daughter Anne, who was left the rest of his property and married Lord North.[1]