What is the proper adjective for the UK?

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.

Solution 1:

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:

Britain, UK
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.

Solution 2:

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.