I love how these two bits of canon seem to go together quite well. If Earth is only fully unified in 2150, it actually makes sense for the Royal Navy to have existed just about up to Enterprise, and possibly after as a historic organisation, what with Enterprise being in the 2150s.
I'm with others in thinking neither the USA or the UK exist as administrative entities at that time, but place names are likely to be kept (such as individual US states, and the constituent countries of the UK, minus NI, which is obviously part of a Unified Ireland before this point).
That being said, there are probably even much fuzzier borders than today. I can imagine several small towns to the very north of England being insistent on being Scottish.