Robert Wilmot was the elder son of Robert Wilmot (died 1738) of Osmaston Hall, and his younger brother was the judge John Eardley Wilmot (1709-1792).
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Wilmot was the second son of Robert Wilmot (1669–1738), of Osmaston Hall, near Derby, and his wife Ursula, who was the daughter of Sir Samuel Marow, Bt, of Berkswell, Warwickshire.