Commonwealth, you say? I thought he was only the Chief Rabbi of the UK? After all, Rabbi Mirvis used to be the Chief Rabbi of Ireland.
The way that I understand it, the UK's Chief Rabbi remains the Chief Rabbi of the entire Commonwealth, the former British Empire. From Canada to New Zealand and back again. I'm not sure how that plays out in rabbinic politics, but my guess is that Rabbi Sacks would supersede because he is just that respected and well-liked. Whether any other Chief Rabbi today would be able to supersede Chief Rabbis of the Commonwealth countries may be a very different question.
Several Americans were considered for the post, which surprised me. It would be like the British Invasion in reverse. With less drugs and music. But still lots of screaming fangirls. And/or it would be the American Revolution Part Deux (Thank Dear Husband for that one).