I think Labour are far more likely to get the NHS updated and performing more efficiently that the Tories, who despite their claims, want it to disappear. We are all paying more for private treament as evidence of this.
As for Australia- you get what you pay for as per private medical insurance- the more you pay in the more you get out, which can be quicker appointment but also access to unneccesary procedures and even surgery. I know this because I have two siblings, one who pays for costly private insurance, and one who doesn't and gets the cheapo level Medicare. They are both elderly and have serious health conditions but my sister gets tested more and has had several surgeries which didn't work, my brother is in reasonable health and doesn't hold with the 'quacks' who have harmed my sister's health by over prescribing drugs that made her very sick.
In Australia it is all about choice. That's what the Tories delude us in to thinking that's what private health care provides, but it is still on top of free-to-access care we have already paid for.
It is true that doctors and nurses are better paid in Australia though, and have better working conditions so I would say private health care benefits the workers more, but not necessarily the patients.
We can have more NHS staff without compromising the NHS- it just needs some commitment, common sense and a bit of vision, I'm not sure we have enough of that though.