The switch in Australia came with decimalisation. We can get a basic history of the switchover from the Reserve Bank of Australia's website.
A big decision concerned the name of the new currency unit.
Views varied widely from traditional labels such as 'pound' and 'dollar' to more distinctively Australian names such as 'Austral'.
A public naming competition seeking suggestions 'with an Australian flavour' added nearly 1,000 names to this list including such exotic suggestions as 'Oz', 'Boomer', 'Roo', 'Kanga', 'Emu', 'Koala', 'Digger', 'Zac', 'Kwid', 'Dinkum' and 'Ming' (the nickname of Prime Minister Menzies).
In June 1963, with no clear consensus having emerged on a name, the Government decided to name the new currency the 'Royal'. Treasurer Harold Holt explained that the Government saw this name as 'emphasising our link with the Crown' and as being 'a dignified word with a pleasing sound.'
Between June and September 1963, the Bank's Note Printing Branch developed a variety of design concepts for the Royal notes.
and
The Royal designs were not completed because of widespread opposition to the name 'Royal' for the new currency.
Just three months after announcing the 'Royal' decision the Government conceded on 19 September 1963 that the name of the currency unit would be the 'Dollar'. This decision won quick and general public approval.
I've had a search for Commonwealth Cabinet discussions about the name Dollar as opposed to other alternatives to Royal because those would be the most likely place to inform us if retaining Pound was seriously considered, but have come up blank.
This basic history can at least demonstrate that there was a public contest, the government initially chose "Royal", and then that it hastily changed to Dollar.
Also I don't know if we have any FX people in the house but why isn't the Canadian dollar considered one of the queen's currencies and quoted as such? Like usually in FX queen's currencies are quoted as GBP/USD, but not the CAD. Always bothered me.
I provided an answer for this question regarding Canada a while ago.
I provided more details, explanation, and citations in that answer, but I summarized it as follows:
The Canadian currency is called a "dollar" for the same reason that the US currency is: it comes from the Spanish dollars or "pieces of eight" that were common in colonial times in North America.