Tax law varies from country to country and depends on (sometimes changing) treaties between countries.
Generally, where you file your tax return depends on your (primary) country of residency. For income earned in the other countries, you may need simply declare it as foreign income, or file a tax return in that country and claim the taxes paid as a deduction on your return for your primary country.
IIRC, when I did it, I filed a return in both countries, but in the US only I declared the AU income also and claimed the AU taxes as a deduction, however for the AU I simply didn't declare my US income in AU. That may have been because I was moving permanently from AU to the US.
(Caveat: My experience is from 1994 when I moved from Sydney to the US)
Note: It is quite likely that rules between AU and the UK (and other British territories) are much more lenient than between AU and US, and certainly than between AU and, say, Saudi Arabia.