The United Kingdom of Great Britain and Northern Ireland consists of England, Scotland, Wales, and Northern Ireland. The Isle of Man and the Channel Islands are British Crown Dependencies but are not part of the UK.
The diner inside is not as seen in the show, but the sign is still there and there's local history behind it. There are framed newspaper clippings and the menu acknowledges the show's three waitresses: Alice, Vera, Flo and the owner: Mel.
Alice's dream was to become a singer but for the time being she got work as a waitress in a greasy spoon, Mel's Diner after her car breaks down in Phoenix. Mel was gruff and demanding and constantly bossing his three waitresses around.