I've heard Ukonian used, and I must say I rather like it, but I don't think it's a fully accepted word yet. British leaves out Northern Ireland.
Wikipedia gives the demonym of the UK as British or Briton.
It's also worth noting that although Great Britain is England, Scotland and Wales, Britain is sometimes used as an abbreviation for the United Kingdom of Great Britain and Northern Ireland.
From the Guardian style guide:
These terms are synonymous: Britain is the official short form of United Kingdom of Great Britain and Northern Ireland. Used as adjectives, therefore, British and UK mean the same. Great Britain, however, refers only to England, Wales and Scotland. Take care not to write Britain when you might mean England and Wales, or just England – for example when referring to the education system.
I don't think it's that uncommon to refer to all of the UK as Britain, and its inhabitants as British. Wikipedia says:
The United Kingdom is often referred to as Britain. British government sources frequently use the term as a short form for the United Kingdom, whilst media style guides generally allow its use but point out that the longer term Great Britain refers only to England, Scotland and Wales.
I'm from Turkey so I can't say that I'm an expert at these kinds of topics, however if I'm not completely mistaken the term "British Isles" is a geographical term meaning Great Britain plus Ireland the island (as in the more poetic Emerald Isle). If we look at the situation from this aspect, using the term British does not exclude the Northern Irish, but in fact it includes not only them but potentially also the Irish (as in Republic of). If you look at it like this, it's like using American for those from the United States, when it could potentially mean anyone from the whole of Americas.
Please note that I'm not criticising the term, in fact I think it's pretty smart to come up with something like that and making it stick. It's ethnically neutral so it does not offend anybody, not even naturalized immigrants. For example in my country some members of minorities no longer want to be called Turkish even though they're citizens of Turkey, they say it's the name of an ethnicity they're not part of. So a term like British is historically very smart for a nation that has many components in its mainland, and many dependencies. I have no idea how it came to use or if it had anything to do with this idea though.
I've heard Ukonian used, and I must say I rather like it, but I don't think it's a fully accepted word yet.
What a bizarre and peculiar word! I have not heard it before. It sounds contrived http://dictionary.cambridge.org/dictionary/british/contrived?q=contrived, made up http://dictionary.cambridge.org/dictionary/british/made-up_2, to me.
British leaves out Northern Ireland.
No, it does not have to. According to this link http://dictionary.cambridge.org/dictionary/british/british_1?q=British, it means of the United Kingdom of Great Britain and Northern Ireland.
There is nothing wrong with using the word British, in this context. Why anyone would think that they need to use another, is bizarre and surprising.