It’s kind of a shitty name to insist upon given our history with Ireland though, isn’t it? Like, regardless of what it was called, we can call the archipelago “the British and Irish Isles” or something if we want to.
Personally I reckon we should call it Maughold’s Isles. “British and Irish Isles” is fine, if a little wordy. “Islands of the North Atlantic” is one I see floated every so often, but it’s miserably generic and even longer. So I suggest we use the patron saint of the Isle of Man. It’s in between Britain and Ireland and technically not part of the UK. Maughold himself was a pirate who tried to play a practical joke on St Patrick, so he’s a bit of a scoundrel, and it’s exactly the kind of silly trivia that we like so much here