Hi, Scotland and Ireland (and Wales) have and always will be part of the British Isles (In spite of some politicians trying to redraw the geological map
) - so they are British, and Ireland was, at the time, under the rule of a British King.
But you are correct, it wasn't the British who won Waterloo, it was Britain and her Allies who won the battle, and Bluchers arrival WAS decisive... but as to who what why where and when things may have been different, better men than I have debated it and still not agreed. Point is Wellington was the commander, and he won it.