British expat in Australia.
You don't have to have private health cover. You can use the public system if you like.
There are free GPs, but they are generally not as good as the charging GPs. You can book an appointment with any GP you like. I can usually get a same day appointment for my paid-for GP, and she is excellent and very thorough. Receptionists are very polite. You are treated as a customer. My GP costs about GBP40 for a normal visit. Blood tests and standard vaccinations etc are free.
You can get treatment in public hospitals if you want, or you can pay for private cover. We have top hospital private cover, costing about GBP 88 per fortnight for a family of 4. It is cheaper if you want more basic cover , or if you earn less money. My fund is a not for profit. There are various out of pocket payments when you use it, but nothing too huge. I can still choose to use the public system if I want. Public hospitals are high quality but longer waiting lists etc.
The health funds cannot refuse to cover you and, after a 1 year waiting period, they have to cover all pre-existing conditions.
Dental is all private. You can pay as you go or get private insurance that covers "extras" like that as well.
Eye tests are free. Glasses you have to pay for.
I'm very happy with the health system here and I don't mind paying as we can afford it and health is a priority. I never had private health cover in the UK so can't really compare the two.