Not everything is covered by the NHS and there are often long waits to be seen.
Of course, it does depend on what you need to use the NHS for.
I'm UK born and raised and have not felt the need to take out private insurance as of yet. However, I've never had any medical issues that require anything more than a prescription or a blood test and I have not had an operation or spent a night in hospital, so I haven't experienced any long waiting times.
I just went to the GP yesterday though, as I've had a cough for several weeks and wanted to get it checked out. I tried to go a couple of weeks ago, but the earliest appointment I could get was 3 weeks out (Aug 7th). However, I know they keep some appointments free for booking on the day, so yesterday morning, I logged in at 7:40 am and managed to get an appointment for 8:50 am that day.
My appointment experience:
08:40 am - Left the house
08:42 am - Arrived at the GP surgery and checked in electronically
09:03 am - I saw the GP
09:08 am - I had been examined and I left the GP's office with a prescription for antibiotics
09:10 am - I put in the prescription with the on-site pharmacy
09:14 am - Dispensed prescription in hand, I got in my car
09:16 am - Got home and took my first dose of antibiotics