As mentioned in the comments, taxation is based on residency and source of income, not just citizenship. The US, as an exception to the rule, taxes its citizens regardless of physical presence rules, but most other countries don't.
So if you live in Australia, you wouldn't worry about the UK taxation even if you're a UK citizen. But it works both ways - you can't get UK tax discounts either.
You're asking about which treaty benefit is applicable to you when it comes to the US tax withholding on US sourced income. Note, withholding doesn't necessarily equate to taxes due.
The tax treaties in question are to resolve issues of double taxation, and are generally based on tax residency. Since you're the tax resident of Australia, the treaty relevant would be the US-Australian tax treaty. To explicitly clarify that, both the treaties (UK, Australia) start with:
Except as specifically provided herein, this Convention is applicable only to persons who are residents of one or both of the Contracting States.
It doesn't matter which passport you present at the immigration.
Will the UK (or anyone else) come chasing me for a chunk of my winnings ?
If the income is taxable in Australia then Australia will, most certainly, as they should, since you'd be committing tax evasion there. The US as well, since you'd be committing tax fraud in the US and depriving Uncle Sam from what is rightfully his. Both might also want a chunk of your freedom, since tax evasion and tax fraud are crimes punishable by jail time in both jurisdictions. All the three countries involved are known to be sharing a lot of information between them specifically directed at identifying financial crimes.
