“Ascension Thursday” is only ever celebrated on a *Thursday *-- It cannot be otherwise. Likewise, “Ascension Sunday” can only be celebrated on a Sunday.
However: There is no celebration by either name in the Church’s calendar. The Church *does *include a celebration named “The Ascension of the Lord” (Sollemnitas Ascensionis Domini), which is ordinarily celebrated on the Thursday falling 40 days after the celebration of the Resurrection, but which Canon Law empowers episcopal conferences to translate (to a Sunday or any other weekday) and/or to suppress the obligation to assist at Mass.
In the US, the episcopal conference seems further to have allowed this to be done according to ecclesiastical provinces, which New York, Newark, Hartford, Boston, Philadelphia, and Omaha* have retained the celebration on Thursday; others have translated it to the following Sunday.
(* The dioceses of: New York, Albany, Brooklyn, Buffalo, Ogdensburg, Rochester, Rockville Centre, Syracuse, Newark, Camden, Metuchen, Paterson, Trenton, Hartford, Bridgeport, Norwich, Providence, Boston, Burlington, Fall River, Manchester, Portland, Springfield, Worcester, Philadelphia, Allentown, Altoona-Johnstown, Erie, Greensburg, Harrisburg, Pittsburgh, Scranton, Omaha, Grand Island, Lincoln)
I cannot speak intelligently about the situation in Australia.