Westcott's year of birth is unknown, but appears to have been between 1752 and 1753, and was likely in 1753. He was the son of a baker in Honiton, Devon, and was baptised on 24 April 1753. He joined the Navy sometime between 1765 and 1768, and by 1768 he was serving as master's mate aboard the frigate . He spent five years aboard "Solebay", rising to the rank of midshipman, and spending time under George Vandeput. He then moved aboard , where he spent the next three years under Samuel Barrington and John Leveson-Gower. He passed his lieutenant's examination on 10 January 1776 and received his promotion to that rank on 6 August 1777, moving aboard . Under her captain, Samuel Granston Goodall, "Valiant" was present at the First Battle of Ushant on 27 July 1778, after which Westcott and "Valiant" joined the fleet under Sir Charles Hardy in 1779. He was present with Vice-Admiral George Darby's fleet when they relieved Gibraltar in April 1781.