If the republic passed in the form presented, i don't think there'd be much difference than the Governors-General we have had. They'll be politically neutral, very bland and drawn from the military and diplomatic corps.
1999-2004 - Sir William Deane - The G-G at the time, acceptable to both sides.
2004-2009 - General Peter Cosgrove - A friend of Howard, hero of East Timor, acceptable to both sides.
2009-2014 - Quentin Bryce - Queenslander, first female appointment, acceptable to both sides.
2014- Currently vacant. Candidates proposed unable to gain 2/3 majority.
A rather plausible list, well done. One minor quibble is that the republic would not have come in until January 1 2001, the 100th anniversary of Federation. So move the dates back two years....
William Deane - 1/1/01-31/12/05
Peter Cosgrove - 1/1/06-31/12/10
Quentin Bryce - 1/1/11-current
Of course, we have a situation in 2011 with a hung parliament...I can see two scenarios there - would Gillard use this opportunity to ease KRudd out of parliament (which would basically need the consent of the Abbott opposition, probably unlikely).....the second scenario is that the monarchist Abbott essentially vetoes each and every nomination Gillard makes, rendering the position vacant for an extended period of time.
That assumes there are no butterflies in the Howard government as a result of a 'yes' vote in 1999.
Edit - After checking out the
Republican constitution it appears that in the event of a stalemate, an outgoing president would continue in office indefinitely until a successor was confirmed.
Section 60 of the constitution states that a committee appointed by parliament makes a report to the PM on suitable candidates. A president is elected by a two-thirds majority of the combined House and Senate voting together. At the time of their election, a president cannot be a member of a political party, or a member of any federal or state parliament. Interestingly though, Section 63 allows a president to appoint a deputy to act in their stead, and this person (or persons) may be current members of parliament.
How I think that would work in practice would be similar to the concept of an Acting PM while the real PM is overseas. President X goes overseas to represent Australia somewhere, and while he is away, appoints Governor Z or Senator Y to mind the shop. It wouldn't work like an American vice-president, as they wouldn't remain in the role and couldn't succeed to the presidency in the event of incapacitation.
If there was no elected president in place on January 1 2001, the acting president would have been the longest-serving state governor to that time, which would have been Sir Guy Green of Tasmania.
Finally, there was a little nugget buried deep in the proposed constitution that read like this (from Schedule II):
"5. The States
A State that has not altered its laws to sever its links with the Crown by the time the office of Governor-General ceases to exist retains its links with the Crown until it has so altered its laws."
So, you could quite easily have a situation where (for instance) the Queensland governor was appointed on the advice of the Queen of the UK, even though Queensland is a part of the republican Commonwealth of Australia.