Portugal

If it doesn't require immediate attention I call my health center. I can get an appointment with my family pshysician in a few days or, if it's more urgent, some other doctor will see me the same day but I'll have to wait there until one is free (can go anywhere between 15mn and 2 hours). I'm lucky though, some health centers suck really bad. The ones in big cities are generally better.

If it's more urgent I call the national health line and they'll A: tell me how to treat it myself B: set up an appointment in my health center (or another if mine is not available) C: send me straight to the closest emergency room.

Wait times in the emergency room depend on the gravity and the hospital. My hospital sucks. Low priority you'll spend there the whole day, easy. 10+ hours. Medium priority you'll wait 4 or 5 hours. High priority about an hour, maybe two. Very high priority (head falling off) you go right in. In good hospitals those times are much lower. In the major city I used to live I never waited more than 2 hours for any priority. I also had surgery there and it was great.

Never paid a cent, I think it goes without saying.