I just had one done on Sunday at 6+3. I would undoubtedly qualify for one on the NHS as I have had three first trimester losses, and I spot constantly, but I prefer private scans for that first one. For starters, they're not in a rush, they let you see the screen (the NHS ones I've had, it's often awkward to see or they don't let you see it until they know what's going on) and you can ask as many detailed questions as you like and they'll answer them. Some NHS sonographers will too, but plenty won't and just want you out the door so they can move onto the next patient.
Plus private scans are often more thorough - e.g. likely position of placenta, actual heartbeat speed (as opposed to "present") and they'll answer questions like "So is the yolk sac too big?" right away - most NHS sonographers don't know about a lot of these things as they're just on rotation or whatever and have no particular interest in early pregnancy, and the nurse practitioners I always saw afterwards didn't know either and would just drone on "we saw a heartbeat.." but I know from bitter experience that there is a lot more to it than that even if a heartbeat is a fabulous start.
I suppose a lot depends on the private clinic you choose - I used to go to one with a highly qualified midwife (£90) and now I've moved I had one done with a proper consultant (£110). Money well spent for sanity!
I certainly don't begrudge the NHS not providing as detailed a service, they don't have the resources to do more, and what they do do is absolutely fine for most people. I'm just one of those people who has been through a lot, learned a lot as a result and so wants to know everything.
Mine was absolutely fine, btw 
- textbook in fact! Even after all my questions 