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.
Recently, the restaurant has become known as Mel's Diner. In the 1960's and 1970's, it used to be called Lester's Diner, then became Pat's Family Restaurant in the 1980's, and finally, as of recent, it's now called Mel's Diner. The name change is appropriate considering it's unique association to the Diner in ALICE.