Re: USA and UK surviving into the Trek era
Why wouldn't countries like the US and UK still exist even in Trek's time? They're simply subdivisions of the larger whole: United Earth.
And United Earth did not cease to exist when the Federation was born (it's simply a Federation member world), so there's no reason it should happen here either. As UE is a member of the Federation, so are the US and UK and other countries members of UE.
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