Australia

• for non-urgent regular things book with my GP in an app (pretty sure that’s the case for most GPs too). usually an appointment within a few days; sometimes he’s booked out for a couple of weeks, but if i don’t feel like waiting i can book anywhere else. when i go to my appointment, tests etc are all free. i go to a private clinic so i pay $75AUD (~$50USD) and the govt reimburses me $45 of that - public clinics are free but i go to a speciality clinic
• for non-urgent sick things (or anything really - scripts etc too; i just use it when i don’t want to go to my GP) we have home doctor service: you book in an app and a doctor will come to your house the same day - free
• for specialists the waiting period can depend on what it is, but i have had a 3mo wait for a specialist before :(… usually you go to your GP, get a referral, see a specialist, then perhaps have surgery if necessary. it can be a months long wait for surgeries which is not great, buuuuut it’s also great to go in for day surgery in the morning, and just leave later that day without paying a cent
• for emergency, unfortunately you can be waiting for a few hours… they triage you so i’m sure if it’s a real issue you wouldn’t keep you waiting but for things like potentially broken bones you can be waiting for up to 3hr… it’s all free
• for ambulance it differs per state but in my state (victoria) they aren’t - it’s ~$1400 for an emergency trip. you can also buy ambulance membership for $53/y and it’s free

pretty much anything where i’ve talked about costs or free you give them your medicare (federal health system for everyone - not just low income etc) details and they bill the govt a set amount for time and materials used. GP clinics etc store it on file so sometimes you can just walk out without talking to anyone