Reed became patriotic over hearing Winston Churchill in Storm Front. Dominic Keating himself is English, as are the actors who played his parents. The novels have operated under the belief that Reed is British, indeed this is the first time in the sixteen years since the character was introduced that I've ever seen anyone question the matter.
Then again, it is certainly worth pondering what "British" means in the Trek 22nd century. There's a whole range of things, from Picard theoretically being French while being so British, to Cambridge badgers apparently being overrun by skunks. Whatever be established about "the Royal Navy", it shouldn't affect our ponderings of United Earth or its military much. Every reference to the Royal Navy predates the year in which United Earth finally earned its name, after all. While many things such as UESPA and UESF explicitly preceded that year by a wide margin, a lot about their status may have changed in 2150 sharp, rendering evidence from before that date moot. Timo Saloniemi
Considering what is happening in the present, what is "British" could change radically before United Earth is formed.
Unless the organization suddenly ceased to exist within the 10-15 years prior to the show, I don't see a reason why it can't. There would still be a need for blue water Navies. It was still very much active when Reed was making his career choice, and his Pops was still begrudgingly learning to accept his decision during the show.
I see no need for any specific organization to act as a bluewater navy. It could be the Royal Navy, it could be the South Korean navy, but much more probably, it would be something like the United Earth Navy. I mean, what possible reason could there be for it not being that? Earth was being reshuffled, had been reshuffled for the past century, and by 2150 something definite was finally done - leaving that half-baked would be silly. When Reed was making his career choice, one option might have been the Royal Navy. But that was not stated. All that was stated was that Reed failed to opt for seaborne service because he got seasick. And opting out of the United Earth Navy after the death of the Royal Navy would be as bad as opting out of the Royal Navy while that organization still existed - dad would still have shrieked like an old Nazi. Timo Saloniemi