@Furries - as a former UK resident of over a decade who has moved back to Australia fairly recently, and lived through the Victorian/Melbourne lockdown, I'm happy to answer particular questions as best as I can.
But, in all honesty, luck was part of it. We had a Premier who didn't give a flying fuck if he was viewed negatively for doing the right thing. He wanted COVID eliminated and took steps to do it - and faced a LOT of criticism from the Prime Minister. Yes, we were in a worse lockdown than the rest of the country, but we had the high cases, the other states didn't so didn't need to have such a strict lockdown.
Our small outbreaks have, I think, been handled faster than NSW. Other states have also gone into complete lockdown when they've had minor outbreaks, but NSW seem to have tried to keep things ticking over, and doing area lockdowns, and as a result have taken longer to get on top of the outbreaks, although in fairness they HAVE got on top of them, and perhaps with less financial impact on their residents than we have had here in Victoria.
Another part is as others have said, the UK is a hub. Flights go through the UK to other parts of the world. Australia is usually the end destination. Closing international borders was easier. Especially as a nearest neighbours like China, Singpore etc also closed borders.
Financially, the impact for the government was less, because we didn't furlough. Instead we had Job Keeper which was a flat $1500 per fortnight, and Job Seeker payments were increased by $550, but this reduced earlier than did JobKeeper. This really benefited the lower income earners, and while giving a safety net to mid level and higher earners the government on the whole probably has a lesser bill at the end of this than the UK government has with their furlough payments.
Generally, we are more spread out, we don't have the same number of high density apartments. The one outbreak in a high rise apartment caused it to be put into very sudden lockdown and noone was allowed in or out of the building at all but medical personnel. Emergency food parcels were organised and delivered to residents.
We were put into a State of Emergency, rules were RULES and not guidelines. Fines were immediately established for breaches, and when it seemed they might not be effective they were increased. When we needed permits to go to work more than 5 km from home you would often pass Police checkpoints where you had to provide ID and the permit.
Those who were home quarantining had random visits by the Health Department and if they weren't at home they had to give a damn good reason or face steep fines.
They checked households who were reported, finding people hiding in cupboards, under beds, etc and fined them. If someone had a party, ALL the attendees were fined not just the party holder.
For or my part, seeing the numbers going down gave me hope and strength to keep going. I'm honestly not sure how I'd have coped if I kept seeing high numbers the way you have in the UK. Being in lockdown for nearly a month with zero numbers was ok, knowing that our restrictions were shortly going to be eased.
My DS's school was brilliant and provided us a laptop for DS to use until the laptop we ordered for him was delivered. The homeschooling provided by my DS's schools was brilliant. The teachers were amazing and so supportive.
And I think the biggest thing was the daily media conferences which the Premier and Health Minister/Chief Health Officer ran every day. This included the question time at the end by journalists - including those who had clear agendas. We weren't reliant on the information that the media organisations released, I and lot of fellow residents watched/listened to the news conferences live. I never felt that information was being kept hidden. The numbers were always given to us.
Contact tracing now traces all contacts of the positive case AND their contacts as well - so close contacts of close contacts. But they were only ever able to do that from our first cases that happened after we had reached zero.