I've had the privilege of experiencing both private and NHS care in the UK (in one instance for the same condition).
NHS-wise, it took about 18 months to go from GP visit about leg pain to surgery with an orthopedist. There was significant waiting (generally 2-3 months) between each diagnostic stage, but my life was only affected by lots of walking (I wasn't disabled, nor was it affecting my day-to-day life other than avoiding walking long distances). Post-op I had about 3 months of physiotherapy on the NHS, which was fine.
Private-wise, I went back for the same condition (needed surgery on both legs). I was still in the pipeline for the NHS to get it done, but since I had insurance through work I pursued that as well. As it turned out, I came up on the NHS waitlist in May but couldn't have surgery then because they wanted a few more tests and then I was travelling on holiday. Private could have scheduled me as early as 2 weeks (likely because the consultant was the same, the only difference was the office and the day of week when he worked private vs NHS). As it turned out, I was scheduled for the private surgery on a Wednesday and could have had the NHS surgery on Thursday (total wait time from the first surgery was about 18 months, though they wouldn't have authorised it within the first 6 months anyway to allow the first one to heal).
The only difference in surgeries was the ambiance. On the NHS I had to come in at 7am and then waited around for most of the day not knowing if I'd even have surgery because they weren't sure if they'd have a bed for me post-surgery (they did end up finding a bed and I had surgery around 5pm). I had a short intro-to-crutches session the following day and was discharged.
Private I still showed up about 7am but was heading in to surgery by 8:30. I had a private hospital room with en-suite, and the day after surgery I had the intro-to-crutches thing plus a session of physio in a pool. I saw the consultant both times the day after surgery.
The only other difference for NHS/private was the access to the consultant. On the NHS I only met the consultant once (the day after surgery), all the other appointments were with his team rather than him. Private I met with him once before surgery, and twice since surgery (private), once at 1 month and once at 3 months (and he'd like to see me again at 9 months). I think on the NHS I saw a member of his team at 3 months and 6 months.
I'm taking advantage of private healthcare because it's offered to me through work for free, but I've been very happy with the NHS. Since moving here I've had 2 eye operations, 2 leg operations, a tonsilectomy, and a diagnosis of a life-long condition. Aside from one GP at my current practice who's a bit off-putting (very firm on the "one appointment, one problem" rule), everyone I've dealt with has been friendly and professional. The key difference I've found with private is that things are both more efficient (if I show up for an appointment at 1pm I'm going to be seen by 1:05, whereas on the NHS I might wait for a while), and more relaxed (they have more time for chit-chat and more "how are you doing overall?" types of questions).