Actually Americans traditionally don't mix Scottish and English under British. I do, but i'm one quarter British Isles heritage, so I don't divide that up. An American Scot who knows they're Scottish would never AFAIK call themselves british. That would be rare in any case.
But, Can you tell apart those Irish phenotypes in most of the UK? Therefore being hald english half scottish is way different than being half italian half finnish, the ethnic mix is only because both countries are not the same (politically) but the people living there are pretty much interchangeable.