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.
This is a similar, more colorful, coffee cup sign from Lester's Diner in Bryan, Ohio. The similarities of this sign and the one shown in the opening credits of ALICE are remarkable. UPDATE: Mel's Diner in Phoenix originally opened in 1963 under the name Lester's Diner, named for it's owner Lester Bammesberg. This is the same owner as the restuarants in both Bryan, OH and Fort Lauderdale, FL. This explains the similarities, as Mel's was a takeoff of the original Lester's.