The "The" in front of USA and UK seems not to be just because the country names are "initials" - I think this pattern is true for many names that read like short descriptions of the constitutional status of the country. (such as "long form" names which start with "Republic of" / "Kingdom of" / "Union of" etc.).

I think many countries have an official "long form" name ("The Kingdom of Liliput") and a "short form" ("Liliput"), which ends up being used almost as if it were the formal name of the country. However, in the case of the USA and the UK, there seem to be no good "short form" options.

USA = "America" is unsatisfactory because there are other Americas (and Frank Lloyd Wright's suggestion "Usonia" never took off).

UK = "England, Scotland, Northern Ireland and Wales" or "Great Britain and Northern Ireland" are not particularly short.