Gangle - In my (limited) experience ? my own and NCT friends ? the first scan is usually at 12 weeks unless there's a problem or a history of problems.
If they offer nuchal scan as standard (which not all hospitals do, so you might want to check) then they'll do it at the 12 week scan - which is therefore usually dating and nuchal in one. 12 weeks is the ideal time for nuchal which needs to happen between 11 and 14 weeks if I'm not much mistaken.
If I were you, then I'd have the scan wherever, but try and swap your care to the hospital you want to give birth in soon after. If all is well there will generally be quite a hefty gap twixt your scan/booking appt anyway, so you might as well make the switch then - you can swap anytime, as Lynette said, but it can be nice to have some continuity of care IMHO.
I agree that since you had no idea of your dates then an earlier scan might have been useful so they didn't inadvertantly miss the window for the nuchal scan (if they do it) but since you've had one privately and your NHS scan date works out perfectly I wouldn't worry too much.
I know it feels very much like you're pretty much ignored by the medical profession in the early months - it's such a momentous thing to happen to you and they don't seem that interested!
FWIW, because I have a long cycle (which I tried to tell my GP about but which they chose to ignore - again!) my '12 week' scan ended up being only 9 weeks and therefore too early for nuchal (which is offered as standard at the hospital I'm with) but they rebooked me at a better time (13 weeks in my case) on the spot, so it wasn't a problem.
But in your case "everything appears to be in order" so I'd try not to worry! In general, the less appintments you have, the better you can assume things are going!? .
Good luck,
db
xx