I had loads of them because DD wasn't a massive mover (or she was, but I didn't feel much) and I had GD. I also have a SIL who had what was clearly a dodgy placenta and their DD was born at 42 weeks at only 4lbs.
I would pay to have one at about 32 weeks if I got pregnant again, just to check things like the cord and the placenta are all functioning well. Apparently that's standard in other parts of Europe.
Baby moves much less thought out this pregnancy than my previous pregnancy. My last baby was very big resulting in traumatic birth. I had nightmares about having baby with birth defects. Baby is transverse I am worried about her being caught in the cord... I spoke to my gp and ask him if it would make sense for me to have late pregnancy scan even if I pay for it. His response was, if I found out it is a large baby what are you going to do? He said don't worry about position we can turn her for you ... essentially said there is no point.
I would recommend talking to your midwife/consultant and getting (forcing) them to listen to your concerns.
My #3 didn't move as much as my others-I went in for monitoring three times after reporting reduced movement. She was also measuring small so the NAH gave me a scan at 34wks which didn't show any problems with placenta or cord. However I was induced after the third episode of reduced movement (at 38wks) and it turns out that the placenta was small and the cord hadn't inserted properly. A scan hadn't shown either of these issues.
My point isn't to scare you it's to stress that a scan alone may not be what you need, especially if you go private-you then have to get the NHS to look at your results. Much better to get the medics to understand your concerns and work with you.