Pushka, you only need a Day 2 Lateral Flow test. You need to purchase this prior to your trip however, and you need to fill out your passenger locator form (PLF). A section of the PLF will require you to put in a reference number from your purchase of the Day 2 test, which you will get when you buy it.
Here are two options for Day 2 tests under 20 GBP (approx AUD 40) - the first one is the easiest and will take you the shortest time to sort out and doesn't require a video call or anything, I've used them a few times:
Fully Vaccinated Arrivals To The UK | Day 2 tests | Qured
You need to buy a test
separately for each pax. Also, you have to make sure you're buying a
Day 2 test, not just any lateral flow test.
Once you've purchased it, you'll get an email with a reference number, which you'll need for your PLF. The link for the PLF can be found here:
Check if there are any COVID-19 rules in place for travel to England from abroad.
www.gov.uk
I know it may seem complicated, but its fairly simple. Buy the test and then do the PLF before you check in (you won't be able to check in without the PLF filled out). Also, make sure you download a copy of the PLF completed or print it out.