Lightman wrote:The current constitutional status of Australia has nothing to do with that of the United Kingdom. Legally speaking, Australia and the United Kingdom happen to share the same monarch.
Beyond that, I don't think Scotland's secession would have any legal bearing on the United Kingdom's existence as a state. The secession of the majority of Ireland does not meet there is no legal continuity between the United Kingdom of Great Britain and Ireland and the United Kingdom of Great Britain and Northern Ireland. It's possible that the United Kingdom would be renamed the United Kingdom of England, Wales, and Northern Ireland, but that wouldn't have any legal meaning.
The United Kingdom remains a popular general name for the entire region, and it's existence or non-existence as a direct political state is irrelevent, as it has come internationally primarily to mean the entire geographical region along with the "common" regional flag of the Union Jack. One can now interpret it almost entirely to be primarily the various military and political alliances the main countries of the region are involved in. This is actually the reason why it would not affect the Australian constitution at all, even if mentioned in the Constitution, it would be interpreted in the modern context of the UK as a political alliance. If the "political state" as it existed at the time is specficially mentioned, it could be easily interpreted as referring to the current British, and Australian, Monarchy and the current British Government, the Goverment of the UK "at the time of the writing of the Constitution".
In it's own special way it has now taken on the simular modern connotations as the word ANZAC, which pretty much just means the Australian-New Zealand political/Army/economic alliance in it's modern usage. Originally it was only the middle one, the last two letters standing for "Army Corps"(Hence Anzac Day this Thursday).
So to answer the question posed in the thread title, NO.