Grant was born at Charing Cross Hospital in Hammersmith, London, the second son of Fynvola Susan MacLean (b. Wickham, Hampshire, 11 October 1933; d. Hounslow, London, July 2001) and Captain James Murray Grant (b. 1929). Grant's grandfather, Colonel James Murray Grant,DSO was decorated for bravery and leadership at Saint-Valery-en-Caux during World War II.