I think the way it works is that if the surgical procedure has a Medicare item number (your surgeon could tell you that), then Medicare will cover at least some of the costs. Your surgeon could probably give you an idea of how much Medicare might cover or you could ring Medicare to ask.
As for private health cover, I think the short answer is that it's best to ask HCF. It depends first of all on the level of cover you have. It might also depend on what hospital you'll be at. Private health insurers can have agreements with different hospitals, so you may need to ask HCF if they have an agreement with the hospital you'll be at.
(In my own case, I had the highest level of hospital cover so I knew I was covered for things like staying overnight in hospital. I also knew that the hospital I was at was at had a good agreement with my private health insurer.)
Things like the anaesthetic and pathology costs can add up too, so it could be worth asking your surgeon and HCF about these costs as well, and to what extent Medicare and private insurance might cover them.
Also, bear in mind that your surgeon may charge a gap fee (i.e. an extra amount you will have to pay, over and above anything that Medicare and private insurance might cover). So if you haven't asked your surgeon about that yet, I definitely would.
Good luck with your queries and your surgery. Hope you have a speedy and smooth recovery