You're confusing the way of paying for a health system with the organisations that deliver the care.
Other European countries have universal health care but different ways of paying. France for example has charges to see a GP, but if you're poor you can claim the money back. Most countries have some kind of mandatory health insurance system with provision for unemployed, students, kids, retired etc.
Then you have the organisations that deliver the care. In France for example, they have a mixture of state, non profit and for profit hospitals. However, there shouldn't be a difference in what you pay between state, non profit etc (with the exception of a few).
How would more private organisations involved help the UK? Well, it entirely depends on which organisations and how you use them. The private sector tends to be good at doing a few simple things lots of times really efficiently. It tends to be bad at complexity. The problem is that the NHS organisation may not be well set up to use the private sector in the right way. However, we can see from other countries that it can work.