Until a couple of years ago I got through life without needing anything more serious than my wisdom teeth taking out in the 80's. They were not cutting through at the right angle, and were threatening to go through some nerves. The waiting list at that time was horrendous and I had to suffer for a year. Eventually I had 3 day stay in hospital in a wee room at the back of the A&E ward, where it was impossible to sleep at night with the arrival of the evenings drunken casualties.

I think a lot of the horror stories of the NHS stem from that time period.
In the last couple of years I've had to attend outpatients very regularly, and the situation has improved vastly.
My main consultant regularly recommends checks and tests at other departments in the hospital. Whenever he recommended eg, scan, x-ray, heart monitor, the appointments usually come through within 2 to 4 weeks. For one of the major tests they got me the appointment within 4 days. All meds that I've needed have been provided free by the hospital's on-site pharmacy
Sometimes I visit an online group for people with similar problems. From there it seems the US members have a lot more worries about the financial burden of their treatment. Some have to get their insurance company to agree to pay before they agree to treatment, some have to share the cost and pay half with their company paying the rest, and it really adds to the stress of their condition.
However, I do think the US has more choice about visiting consultants in other areas than we do in UK. The US members frequently recommend specialists and experts to each other. With the UK NHS you are more or less allotted to your local hospital, and can't decide eg. that you prefer the clinic 50 miles away.
For routine GP appointments there are 5 doctors in the local practice. I can get an appointment within 1-4 days, but sometimes it can take a week if I'm being fussy and hold out to get a particular doctor.
I hope that's the sort of info you're looking for.