What is a non-trivial procedure? In fact the issue would be trivial (or unjustified) procedures, since anything life threatening would obviously have priority over anything else.

In northern Italy I have seen two "simple" (easily removed surgically) cancer cases going through all diagnostic steps in less than 2 months (including ultrasound, biopsies, CT scans), followed by surgery and/or radiotherapy, all at absolutely zero cost for the patient.

If you need heart surgery (valve replacement, coronary stents, stuff like that), your GP can get you an ambulance straight to the ICU, and you can stay there for free until everything is ready for surgery, maybe in a couple days.