[3] Margaret was the second daughter of Moses and Mary Baker, of the Parish of St Christopher le Stocks, in the City of London.

[5][2]: 95 [b] Francis became a director of the Amicable Society for a Perpetual Assurance Office by 1775,[7][8][9]: 231  and on 17 July 1787, The Right Honourable Henry Dundas appointed him Cashier of the Navy.

[12]: 48  Cooke served aboard Eagle, the flagship of the North American Station, during the next three years, seeing extensive action along the eastern seaboard.

Notable among these actions were the naval operations around the Battle of Rhode Island in 1778, when Eagle was closely engaged with American units ashore.

"[2]: 95  On 21 January 1779, Cooke was promoted to lieutenant and joined HMS Superb in the East Indies under Sir Edward Hughes, but was forced to take a leave of absence due to ill-health.

[2]: 95 Cooke returned to England and then went to France to spend a year studying, before rejoining the navy in 1782 with an appointment to the 90-gun HMS Duke under Captain Alan Gardner.

He remained with Gardner following the signing of the Treaty of Paris in 1783, bringing an end to the American War of Independence, and served as his first-lieutenant aboard his next command, the 50-gun HMS Europa.

[2]: 95  He had recovered sufficiently by the time of the Spanish Armament in 1790 to be able to take up an appointment from his old patron, Sir Alexander Hood, to be third-lieutenant of his flagship, the 90-gun HMS London.

[2]: 95 With the outbreak of the French Revolutionary Wars in February 1793, Cooke rejoined Hood and became first-lieutenant of his new flagship, the 100-gun HMS Royal George, part of the Channel Fleet.

[15]: 80  The French ships attempted to escape into Brest, but were hunted down by the British, who forced the surrender of Résistance and Constance in turn after successive short engagements.

Cooke attempted to assist Admiral John Colpoys at the mutiny's outbreak, and was ordered ashore by his crew when he tried to return to his ship.

Cooke was tactfully removed from command by the Admiralty following the mutiny, although he was returned to service two years later aboard the new frigate HMS Amethyst in preparation for the Anglo-Russian invasion of the Batavian Republic.

[18] During 1801, Cooke participated in the capture of the Dédaigneuse off Cape Finisterre, helping Samuel Hood Linzee and Richard King chase her down on 26 January.

In May, after the large combined French and Spanish fleet, under Admiral Pierre-Charles Villeneuve escaped from Toulon, beginning the Trafalgar campaign, Cooke was ordered to join a flying squadron under Vice-Admiral Cuthbert Collingwood.

The squadron arrived off Cádiz on 9 June and Collingwood detached Bellerophon and three other ships to blockade Cartagena under Rear-Admiral Sir Richard Bickerton.

Cooke took the unusual step of informing his first lieutenant William Pryce Cumby and his master Edward Overton of Nelson's orders, in case he should be killed.

Cumby noted with surprise that Cooke was still wearing his uniform coat, which sported epaulettes that marked him out as the ship's captain to French snipers.

[21]: 52  Cumby's charge cleared the French from the deck of Bellerophon, and he found Cooke dead on the quarterdeck, two musket balls lodged in his chest.

"[12]: 48 Cumby took charge of the battered Bellerophon, directing her fire into Aigle and ultimately forcing the French ship's surrender after the arrival of other British vessels.

Tributes from fellow officers were also forthcoming, including from the future explorer John Franklin, who had served on Bellerophon at Trafalgar and had said of Cooke that he was "very gentlemanly and active.