Earlier quoted context omitted.
In Australia at least you can claim a tax rebate if you have private health insurance. I imagine the UK would have a similar system.
In the UK, private insurance usually requires you to use the public health system as a first stage, and only fall back to the private one for things not covered by the NHS. So you don't get a tax rebate because you are still relying on the public system.
To be clear though, I'm fine with this - the NHS is for everyone and it should be funded for everyone.