Canada and Australia used to be Dominions, but that's not fashionable any more; sometime in the early 80s Canada dropped the "Dominion of" part from its name. We are still "constitutional monarchies" (not kingdoms, that's sexist) and "parliamentary democracies", for all the good that does us.

The only people I know who say "UK" are frequent travellers, especially people who work in aviation, and recent immigrants referring to visits back home. Forty years ago, the BBC and CBC co-operated on a radio show called "Postmark UK" for ex-patriate Brits, featuring musical requests; that's where I first heard the term, although my father was brought up in England he never used it. My husband does, though; his family emigrated after the war. Perhaps it's a post-war thing.