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As shown by Wikipedia, Australian citizens can enter the Schengen area (including Portugal) and stay for up to 90 days out of any 180.
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In Portugal, being a British citizen gives you some extra rights (you can work without permit, stay there longer than 90 days, use the “EU passport” lane at the border and wouldn't need entry and exit stamps) but you can also enter easily with your Australian passport (no visa needed for a short stay) so nothing to worry about.
In the UK, since you are a citizen it's generally a good idea to show the local passport but it's not mandatory (unlike in e.g. the US) and many dual citizen do enter on their “other” passport without difficulty, see Do I have to enter the UK on my UK passport? Here as well, Australians citizens can visit without a visa.
In both cases, not having a British passport might prevent you to use your rights as British citizen but for a short visit it should not create any problems.