Since the UK was a member of the EU, I'm assuming the EU was responsible for any kind of trade arrangements between the UK and Australia, but I'm not entirely sure. Does the UK have an agreement with Australia for trade and if so, when did this agreement take place? Many thanks.
No, the UK does not have a trade agreement with Australia.
Since it is still a member of both the European Union's Single Market and Customs Union (EU SM&CU), it is covered by all of the EU's trade deals. However, the full EU-Australia trade deal is, at the time of writing, still being negotiated.
Should the UK decide to go ahead and become the first country in history to impose economic sanctions on itself, withdrawing from EU SM&CU, then its own home-baked trade agreements will come into force. The embarrassingly unimpressive list of trade agreements that the UK has negotiated is listed here.