Australia does not have any formal requirements around identification for flying on domestic flights, and there are no ID checks as you go through security as there are in many other countries (in fact, you can still enter most Australian airports "secure" areas even without a boarding pass).
Most airlines do require you to have identification, although it is often not checked, especially if you are not checking bags. I've never seen any listings of what identification is acceptable, but I would not expect you to have any problems when using a UK drivers license.
The one exception to this is domestic flights leaving from an international terminal. There are very few of these so it's unlikely you'll be on one, but these do have more strict ID requirements.