Re: USA and UK surviving into the Trek era
. . .Picard is from 'France' as is stated in the show....however that does not mean 'France' in the modern sense. It is simply a region, a geographical area. NOT an independent, autonomous body. . .
This is how I've always interpreted it, since the TOS days. The names are just easy to use descriptive references for the characters (and us) regarding former political boundaries within the one-world government.
Dr. Howard, Dr. Fine, Dr. Howard: For Duty and Humanity!
--Men in Black, 1934.