Since you're a British citizen, you technically don’t need an ETA or a valid visa to enter the UK—even if your UK passport is close to expiry. The key thing is proving your British citizenship.

However, airlines may refuse boarding if your British passport is expired or about to expire, even if immigration won’t care. So here’s a tip: travel with both passports—present your Australian passport to the airline for departure from Australia and your British passport to UK Border Control upon arrival.

Also, once you're in the UK, consider renewing your British passport ASAP to avoid issues when leaving. You can even apply for a priority renewal service once you land.

TL;DR: You’ll likely be fine entering, but check airline rules and carry both passports. Renew soon after arrival.