I've had two private scans so far and one NHS. We're moving to the US tomorrow where all care will be private (yippee!), but if my scan experiences are anything to go by, then it's worth every penny.
The first pricate scan was at 8wks, so earlier than the NHS provides. Not medically necessary, but great in terms of my sanity (I had major abdominal surgery last year so having a consultant tell me that this wouldn't affect anything/my ability to give birth naturally/increase any risks this early in the pregnancy was great).
I then had an NHS scan at 12wks which was ok. Short (4-5mins max including putting on/wiping off gel) and sweet and arguably a bit superficial (she noted that there 'was' a heartbeat, not the rate etc) but did the job. My chat with the consultant after though was great: she was running very late but great. My DS risk after this came back as relatively high (due to bloods, not nuchal measurement) so we went private for a reassurance scan.
It cost £180 but I'd have paid double. They saw us 4 hours after I called for an appointment, saw us on time, took 30mins to look at every little part of the baby they could, talked us through everything, gave us time for questions etc. The bloods were also ready by the end of the scan so we got instant results (which were almost 30 times 'better' than the NHS ones). I felt so much more confident after this.
I also had private treatment last year for a serious illness, and if obstetric care is anything like the care I received, then it will be worth it. Personal attention (every single person at the clinic knew DD's name, the catering staff remember I liked fizzy water rather than still etc), being able to call up at any point for advice etc.
If you go private you'll likely see the same medical professionals as you would if you were NHS, but you'll see them more often and have more contact time with them (and, in my experience, their mobile phone numbers).
The NHS will do the job, often remarkably well (not NHS bashing at all - they saved DD's life twice during her birth), but going private offers the 'extra', and in my opinion is worth it for the reassurance this provides.