Scots, Welsh and Irish (both types) are all "Brits" by being residents of the British Isles, so there is no distinction to be made anyway.
The English are "Brit" as well.
The British Isles covers the whole thing - the UK is everything excluding the Republic of Ireland (the larger southern part, formerly Eire), then the individual countries of England, Ireland, Scotland and Wales.
Personally, I put "British" as nationality, unless it's not an available option - whereupon I revert to English.
Which leads to the fact that I've never once met one single person who would consider "European" a viable answer. Even in continental Europe.
It's like how everyone in Canada, the US, Panama, Chile, Brazil (etc.) are all Americans.