I know nothing about Greek healthcare but after reading your post I looked it up on Wikipedia.  It says, and I have no way of knowing how correct this may be, that technically illegal healthcare clinics have sprung up following the austerity measures in recent years.  They are supported by private donations.  I expect you have already looked up this question on the internet and almost certainly have a better idea of what actually happens in practice.    I don't know where else you might go for information.  My best wishes.

Hi

Like your family member I too live. In greece.  Over the past few months I have had numerous hospital stays, scans, tests and medical examinations.  There is a health service here and I have found them very thorough, helpful and quick. By using my AMKA (social security number) obtained from my local council office. You just need your residency card (should have one if they own a vehicle) plus your passport. It's free of charge and Only took ten minutes to organise. This enables you to get cheaper prescriptions, and health benefits.  I have also  used several private  laboratories for blood tests and ct scans but these are so cheap compared to England that it's never been an issue.

I live on an island and all the facilities have been open to me just like the nhs would be if I'd be a uk resident.

My experience of waiting times and procedures in England is far worse than here. I've had appointments made immediately here for me so please be encouraged and not discouraged.

I hope this helps and I wish your family member every good wish for a speedy recovery.