How can I control for trade between UK and Australia in my difference-in-difference trade model?

I can’t imagine what dummy could you use there. Here trade is your dependent variable if you would use dummy you would have to switch to some non-linear probabilistic model such as logit or probit.

Most of the time I see papers using just exports from country i to j as proxy for bilateral trade. But there are multiple ways how you could in principle measure it, and it’s not easy to provide some short review of all possible ways. For example, more sophisticated metric would be to measure it in terms of value added like here. It would really depend on exact situation, I would maybe recommend starting with the exports from UK to AU for UK and vice versa for AU and then move onto more fancy measures, it might be good idea to do extensive robustness checks using different possible metrics.