There is still out-of-network healthcare (i.e. specific services or entire healthcare providers not covered by single payer) in many countries with universal healthcare. But it is usually clear which is which.
Can you provide links?
I've personally used the healthcare systems in Australia and Canada for two decades each, and also for a short time in the UK. I've never heard of this.
The treatment provided will be similar to the NHS, but with less waiting (if relevant) and nicer facilities, such as private rooms rather than shared wards in hospital.