Australia has enough tanks. In case the Queen want to dissolve the government they can just tweet "Nah" and continue as usual. What is the United Kingdom going to do? Invade? Ask Canada and New Zeeland to make a joint force?
(Officially breaking will cause some problems, so it depends a lot on the internal situation and how much popular support the government has.)
Rereading the wiki article, it seems Australia resembles England before the House of Lords was broken in the 1905-1909 time period - the upper house can influence money bills. It would seem that this was a case of the Governor General involving powers of the crown that it has but should not use. The gg was Australian.
Perhaps this is (another) example of the difficulties of running a government composed of more than one centre of power.
(Officially breaking will cause some problems, so it depends a lot on the internal situation and how much popular support the government has.)