Hmmm - if we have Rosebud do we also have a Citizen Kane somewhere?

or should you only say, "I immigrated to Australia," once you have emigrated
I believe so, MG.
Basically it depends which end you're looking from. As far as people from your motherland are concerned, you will always have emigrated. As far as people from your new country are concerned, you will always have immigrated.

Personal opinion, Rosebud - neither word feels quite right for a significant change of mental state. Bill's suggestion of migrate, although it feels better to start with, implies quite a long process, and also an eventual return. Perhaps not what you want either.

Having said which, I've no better suggestions!

However, I'm reminded of a wise friend's comment:
A change in attitude refreshes the parts a change in lifestyle cannot reach