Actually Australia in practical terms has been independent for over 100 years. We have a Governor General which is the Queen's representative but it is formality. Our conservative party is basically a supporter of the monarchy and the Labor party is very anti the monarchy.
In real terms there is not really much difference between the political systems in America, Australia and Britain. I think most of the differences between the countries are a result of area, population distribution, climate and so on. All three are based on winning areas or electorates as opposed to popular vote. I guess in theory a Prime Minister would have more power than a president (as in POTUS system) because the Prime Minister is in parliament and can vote.