A question of UK nationality...

Ok, I hope someone can satisfy my curiosity on this point following one of those down-the-pub conversations that never came to a resolution.
Given that Britain is England, Scotland, Wales and the UK is Britain and Northern Ireland (as noted on all our UK passports!), is there a term to describe a person from anywhere in the UK (UKish?). British would appear to only cover a subsection.
Pete
Given that Britain is England, Scotland, Wales and the UK is Britain and Northern Ireland (as noted on all our UK passports!), is there a term to describe a person from anywhere in the UK (UKish?). British would appear to only cover a subsection.
Pete