How many private hospitals do you know in Australia that actually have an emergency department? - Most that I have come across simply direct those patients to a public hospital, in order maximise their profit from outpatient procedures.
There are plenty. Private hospitals aren't entirely profit-making enterprises in Australia... there are actually a lot of nonprofit private hospitals that are run by religious groups and the like (i.e. Sydney Adventist Hospital). I did some rotations at Sydney Adventist, and they seem to think that they're a public hospital... they have an ED, an ICU, a cardiac ICU, a labor/delivery ward, a clinical school (for med students, nursing students, and the others), several general wards, etc. Starting this year, they even have interns and residents, and they've had registrars for a few years.
I just thought of one other big difference - healthcare guidelines in Australia are less penny-pinching than in the UK. I don't know about daily practice in the UK, but whenever I look at NHS guidelines, I'm always baffled by how many outdated drugs that they use just because of cost. In Australia, they still avoid spending money unnecessarily (like in the US), but if the patient is likely to benefit from a more expensive treatment, they'll always get it.
On the flip side of that, basic healthcare isn't 100% free of charge like it is in the UK. Aussies have to pay up to $34 to fill a prescription... if the drug costs more than $34, the government takes care of the rest (whether it's $35 or $35,000). Also, Medicare usually doesn't pay enough to cover a specialist visit... they might pay the specialist $75 for the visit, but the specialist will often set his fee at, say, $125... and the other $50 comes out of the patient's pocket (or their private insurance, if they have it).
But ED visits are always free (both the consultation and the treatments), so that probably wouldn't affect you either. At Sydney Adventist, ED visits are $300, and the rest is covered by Medicare.