Seymour was the eldest son of Vice-Admiral Lord Hugh Seymour and Anna Horatia Waldegrave (a daughter of James Waldegrave, 2nd Earl Waldegrave) and joined the Royal Navy in October 1797. He was assigned to the Royal yacht HMY "Princess Augusta" and then transferred to the third-rate HMS "Sans Pareil" in the Channel Squadron in March 1798 and to the second-rate HMS "Prince of Wales" in the West Indies later that year. He was present when the Batavian Republic surrendered Suriname to British forces in August 1799 during the French Revolutionary Wars and, having been promoted to midshipman, transferred to the fifth-rate HMS "Acasta" early in 1800. He joined the fifth-rate HMS "Endymion" in 1802 and then transferred to the first-rate HMS Victory, flagship of the Mediterranean Squadron, in 1803, to the fourth-rate HMS "Madras" in February 1804 and, having been promoted to lieutenant on 12 October 1804, to the third-rate HMS "Donegal" later that month. In HMS "Donegal" he took part in the pursuit of the French Fleet, under the command of Admiral Pierre-Charles Villeneuve, to the West Indies and back in Summer 1805 during the Napoleonic Wars before seeing action at the capture of the Spanish 100-gun "Rayo" in October 1805.