The thing is when you say the British government or British PM the focus is on the government and the PM or whatever you’re referring to, so it’s correct to use British in this context. Everyone would know that it includes NI (at least in the context of government or the PM).
However, you shouldn’t use British to describe anything specific in NI as they would hate that.
My older relatives call themselves British or Northern Irish. If you ever visit Northern Ireland you should be wary about calling people Irish. The reverse is the case in Scotland and Wales, of course. No British there.
If you worded the question carefully you’d be OK. “So, are you British or Irish?” wouldn’t go down well with anyone. You might find yourself nailed down in a car park.
@BiggusDickus, I think @Gain just picked NI on a whim and didn’t fully understand the politics. A lot of people lost their lives just so they can be called British or Irish, and when you have risked your life for something, you don’t give it up easily.
I’ve been seeing the phrase “going forward” with increasing frequency lately, instead of phrases like “in the future”, “from now on”, “henceforth”, etc.