I was born and raised in the UK and in 25 years, I have never had a problem with everyday healthcare needs such as doctors appointments or prescriptions etc. If you are under 18, all appointments/treatments/prescriptions are free (including eye tests, glasses, dental check-ups, fillings etc.) and once you are earning, you contribute to the NHS costs by way of taxes. When you reach the age of 60 (or 65, I can't remember which), prescriptions are all free again.
As an adult, all doctors appointments are free upfront, the standard charge for a prescription is £7.10 (about $11) and certain prescription items are free (e.g. birth control). Dental check-ups are about £16 and eye tests are £20 (or free when they have vouchers and offers on), but you do have to pay for the glasses if you want nice ones! I have never had a problem with doctors or treatments (and the longest appointment I've ever had was 15 minutes, so it doesn't take up a lot of my time), although I have never had any long-term problems or needed surgery.
However I did end up in the emergency room when I was 12, after breaking my leg. It was a Saturday night, but I was seen, x-rayed, put in plaster and on my way back home within about an hour and a half. They gave me crutches and medication and later on, a foot support for my cast... and the entire process - the emergency room treatment, medication and crutches cost exactly nothing!
In terms of waiting lists and things, my gran was told in September that she needed a hip replacement and she had a choice: she could go private to get it done quickly... and pay about £10,000 ($15,000) for the privilege, or she could wait 12 weeks and get it done for free on the NHS. The waiting list wasn't too long so she decided to wait the 12 weeks and her operation should be done just after Christmas, by one of the best surgeons in the area

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On the other hand, my brother, who is a gymnast (recently competed in the British Championships for about the 6th time), has needed several surgeries over the last few years, including one major operation on his shoulder. Because he is a UK athlete, he pays a few pounds a month for extra sports insurance and so can get bumped to near the front of the line for surgeries... he hasn't had to wait longer than a few days to a few weeks for appointments with his specialist and for his operations

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