In fact Ugg boots did originally come from Australia, or possibly New Zealand. It's mists of time stuff.
An American company nicked the name and now have a copyright on it.
Ugg Australia is still very much alive and kicking and I have bought boots (from the seconds shop) when I've been to Oz. They use Ugg Australia and I think use lower case letters as opposed to upper case, which makes the whole thing legal, though totally confusing.
I think it was 2005 when I was asked to sign a petition, in Australia, to get Ugg restored to the Land Down Under.
Darn, no success.