He was commissioned lieutenant on 24 June 1762 and saw duty aboard the Pearl 32, Captain Charles Saxton, being her senior lieutenant when she was paid off in 1766 after service on the Newfoundland station.
During the winter of 1776-7 Watt was a junior lieutenant aboard the Eagle 64, Captain Henry Duncan, flagship to Vice-Admiral Lord Howe in North American waters, and as the first lieutenant of the Roebuck 44, Captain Andrew Snape Hamond, he participated in the Philadelphia Campaign from August – November 1777. Having been placed in temporary command of the captured American frigate Delaware 28 by Captain Hamond with a scratch crew, he destroyed a number of rebel privateers. Upon Vice-Admiral Lord Howe confirming the appointment he was promoted commander with effect from 23 November, although he had left that vessel by 22 April of the following year when Captain Christopher Mason was posted into her.
On 2 July 1778 Watt joined the fireship Sulphur 8, commanding her in Lord Howe’s fleet in the defence of New York during July and being present at the action with the French fleet off Rhode Island in August. He left her in the following month and returned to England.
From 13 September 1779 until 19 April 1780, he reportedly commanded the hired vessel Worsley 32, although as early as January in the latter year it had been decided that she was unfit for naval service. In the early summer he was commanding the Garland 24 in home waters, presumably on a temporary basis for Captain John Stanhope, and on 15 June he arrived with this vessel at Stromness in the Orkney Islands after a week-long passage from the Nore with a convoy of Hudson Bay ships. On 1 September Watt departed Stromness in command of the Pegasus 28 with a couple of ships for Amsterdam, and on 21 November this vessel entered the Yarmouth Roads having ten days earlier, in company with the Cleopatra 32, Captain Hon. George Murray, taken the Dunkirk privateer Comtesse de Provence 18. Watt left the Pegasus shortly afterwards in favour of Captain Stanhope.
He was posted captain on 9 May 1781 and recommissioned the Sultan 74, sailing for India from Portsmouth on 11 June with an East Indies convoy, although after putting into Torbay a week later these ships were further delayed when two French 74-gun ships were discovered just four hours after the convoy had set sail. They eventually put out from Plymouth several days later.
By the time the Sultan joined the East Indies Fleet in March 1782 she was ridden with scurvy. Watt fought at the Battle of Providien on 12 April where nine of his men were men wounded, and at the Battle of Negapatam on 6 July where he lost sixteen men killed and twenty-one wounded. During the action a French 64 lowered her colours to the Sultan but later re-hoisted them and rejoined her consorts. Watt was sent by Vice-Admiral Sir Edward Hughes Hughes under a flag of truce to claim the 64 as a prize, but this demand was rejected by Commodore Suffren.
Captain Watt died as a result of injuries incurred at the Battle of Trincomale on 3 September 1782.