The UK has a phenomenal amount of dialects, they're all British though and even a Londoner can tell the difference between a Swedish accent and a British one.
British accent != accent from the UK. I doubt anyone from Northern Ireland would asociate with being thier accent called British, from either side of the political divide; and I suspect more than a few of our Scottish bretheren would object rightly or wrongly.
But to your point - a Scottish accent and a Swedish one are distinctly different.
Scotland and Shetland are both part of the British Isles, a geographic term. Scotland itself is actually part of Great Britain, an island which contain England, Wales and Scotland. Although admittedly Shetland is not.
The term British is rather malleable but check the first entry in wikipedia:
Yes, common usage of the term British includes the whole of the UK.
The usual problem people from Scotland and Ireland have are being called English, not British and Scots are British anyway and I never mentioned the Irish.
Finally riddle me this, what should I have used instead of British to refer to the people of the UK? Which is where I'm from I might add. As there exists no other term other than British.
From a bit of youtubing and googling the few clips I've heard are instantly recognizable as scottish, with a few words you don't understand.
http://www.bbc.co.uk/news/uk-scotland-north-east-orkney-shet...
http://www.youtube.com/watch?v=Ij60XUvWjZY&feature=playe...
http://www.youtube.com/watch?v=QVxNxnXo7-c&playnext=1...
http://www.youtube.com/watch?v=cx5MgoL5tQ8
The UK has a phenomenal amount of dialects, they're all British though and even a Londoner can tell the difference between a Swedish accent and a British one.