HMS Enterprize (or HMS Enterprise)—the first known vessel to bear that name—was a captured French vessel that would later serve in the Royal Navy. Along with the HMS Victory the Enterprise fought in the Battle of Trafalgar under the overall command of Horatio Nelson.
In the late 18th century, one of the most infamous mutinies in history took place the Royal Navy vessel Bounty. Doctor Leonard McCoy later remembered this event by naming the Klingon bird of prey they had captured in their unauthorized mission to rescue Captain Spock HMS Bounty.(TOS movie, novelization & comic adaptation: The Voyage Home)
The Royal Navy continued to exist into the 22nd century, when several members of Malcolm Reed's family served in the Royal Navy. Reed's great uncle served in the navy on board the HMS Clement, and died saving members of the crew. Reed's father Stuart Reed served in the navy, and was disappointed when his son chose the United Earth Starfleet over the Royal Navy. Reed's grandfather was an ordnance officer in the Royal Navy. (ENT episodes: "Silent Enemy", "Minefield", ST website: StarTrek.com)