The term The Royal Navy without qualifications was used on Earth, even by non-Britons, for the navy of the then United Kingdom, which was for centuries the world's most powerful navy and was still a formidable force in the 22nd century.
As an island nation, England, and later the United Kingdom had navies from earliest times. In the 16th century Henry VIII developed a powerful Navy Royal. With the restoration of Charles II in the late 17th century the structure of a professional armed service had been established.
After various wars with France and Spain in the 17th and 18th centuries, by 1815 Britain, through the Royal Navy, was unchallenged mistress of the oceans. It was during this period that a captured ship was reflagged as British and named HMS Enterprize, the first battleship to be named Enterprise.
The Royal Navy also carried out a major programme of scientific surveying during the 18th century. For example, Captain James Cook mapped out New Zealand and the east coast of Australia, and discovered Hawaii.
The Royal Navy either survived the dissolution of all world powers during the post-atomic horror, or was reforged afterwards. Either way, Malcolm Reed's father and grand-uncle served in the Royal Navy in the late 21st or early 22nd century. (ENT: "Silent Enemy", "Shuttlepod One", "Minefield", "Affliction")
There is no canon evidence of the existence of the national navies beyond the unification of Earth in the 2150s.