In the UK, a 15m follow-up telephone appointment with a private consultant is ~£100-£150. An in person 30m appointment varies between £100-£300 depending on specialism and location. If they decide to prescribe something there is no extra charge. They will write to your GP with the outcome, for which there is also no extra charge.

As regards blood tests and the like, the cost depends on the type of test, but could be anywhere from £25-£300.

I do agree with the GP about a 2-tier system evolving in the EU, at least from my perspective in the UK and what I hear from colleagues around the EU; if you have something easily diagnosed and treated, the state health service is literally a life saver, even if you have to wait several months to be seen. But don't expect your doctors to be experts - your endocrinologist will likely no nothing other than diabetes etc.

Shoulder MR scan in Austria: €235 including 10% VAT

You can usually get 80% back from public healthcare insurance. You can do it all via public healthcare, but wait time is usually several weeks.