Billionaires don't pay any tax! But in our globalised world it's hard to do much about that.
Already More than 25% of all income tax revenue is paid by the top 1% of taxpayers, i.e. taxpayers with the highest incomes and 90% of all income tax revenue is paid by the top 50% -
https://en.wikipedia.org/wiki/Taxation_in_the_United_Kingdom so high earners aren't low hanging fruit ready to pick.
Either tax needs increase or people need to pay directly, the current model of the NHS isn't sustainable. I'd look to other European systems that have a far higher satisfaction rate than the NHS for reforms, but people need to get over it always being free at the point of use. Sure poor people shouldn't have to pay, but I don't see anything wrong with those that can afford to contribute.