Bourne, in his earlier days apparently a merchant and shipowner, served in the Parliamentary Army during the civil war, and on the remodelling of the fleet after William Batten's secession, having then the rank of major, was appointed to the command of the Speaker, a ship of the second-rate. As captain of the Speaker he was for two years commander-in-chief on the coast of Scotland, and in September 1651 carried the Scottish records, regalia, and insignia taken in Stirling Castle to London, for which services he afterwards received a gold medal of the value of £60.
In 1652 he was captain of the Andrew, and in May was senior officer in The Downs, wearing a flag by special authority from Blake, when, on the 18th, the Dutch fleet under Maarten Tromp anchored off Dover. It was thus Bourne who sent, both to the council of state and to Blake, the intimation of Tromp's presence on the coast, and who commanded that division of the fleet which had so important a share in the Battle of Goodwin Sands.
Without knowledge of the battle, the council had already on the 19th appointed Bourne rear-admiral of the fleet, a rank which he held during the whole of that year, and commanded in the third post in the battle near the Kentish Knock on 28 September. But after the rude check sustained by Blake off Dungeness on 30 November, it was found necessary to have some well-skilled and trustworthy man as commissioner on shore to superintend and push forward the equipment and manning of the fleets. To this office Bourne was appointed, and he continued to hold and exercise it not only during the rest of the Dutch war, but to the end of The Protectorate. In this work he was indefatigable, and in a memorial to the admiralty, 18 September 1653, claimed, by his special knowledge, to have saved hundreds of pounds in buying masts and deals; from which we may perhaps assume that he had formerly been engaged in the Baltic trade. Nor was he backward in representing his merits to the admiralty; and although he wrote on 13 October 1653, that his modesty did not suit the present age, it did not prevent him from quaintly urging his claims both to pecuniary reward and to honourable distinction. This last, he says, 13 April 1653, "would give some countenance and quicken the work. I ask for the sake of the service, for I am past such toys as to be tickled with a feather".
After the Restoration, being unwilling to accept the new order of things, he emigrated to America; the last that is known of him is the pass permitting him "to transport himself and family into any of the plantations" (May 1662). On 3 April 1689 the secretary of the admiralty wrote to a Major Bourne in Abchurch Lane, desiring him to attend the admiralty, who wished "to discourse him about some business relating to their majesties' service"; and on 28 June 1690 a Nehemiah Bourne was appointed captain of the Monmouth (Admiralty Minutes). It's not known for certain if this was the same person. If so, he apparently refused to take up the appointment because of his advanced age or died before he was able to, for on 9 July another captain was appointed in his stead.