What’s the difference between UK:A, UK Athletics, and England, Scotland, Wales, Northern Ireland Athletics?

UK Athletics is the national governing body for athletics in the UK. UK: Athletics produces the rule book, and administers race licence/permits.
Races (and their organising clubs) need to be affiliated for insurance purposes etc. Affiliation is to a National Association ie England, Scotland, Wales, Northern Ireland Athletics, not UK: Athletics. In effect races affiliate to UK: Athletics via one of the national association. The National Associations are responsible for their respective geographic areas. They operate more on and a grass roots level, where as UK: Athletics operates more on a national /international level.