I think the main issue you're going to face going private is that you will, unless they opened something up in Edinburgh - it's been nearly a year since we left, need to go to England or Wales for the birth.
As prettybird pointed out, there may not be any private hospitals that handle delivery in Scotland.
Certainly possible, but you might want to do a search on this board about private hospitals in, say, London. There are many who have gone down this route, but you may have some logistical costs which are not covered by insurance.
Also, some hospitals, depending on the trust, will not treat you there antenatally if you are going elsewhere for the delivery.
Your other option is to return to the US for delivery.
I know a couple of people who did this as they were not comfortable with how maternity care and delivery are handled here in the UK, even privately.
During the preganany with a second child, you don't get seen nearly as often as with the first unless you develop a complication.
Leonie is right, OBs don't get involved here at all unless there is a problem or complication like diabetes or hypertension or a problem occurs during delivery.
And then, you don't get to chose the OB.
I had to have a forceps delivery with DD1 and the OB consult was just whoever was there. She was fab, but I wouldn't know her from Adam and it wasn't a matter of choice.
This time round, I only saw an OB to get a referreal for my antenatal depression. All she did was talk to me about my moods and past history of post-natal and ante-natal depression and make the referral. I still have all my care with the midwives because there is nothing physically wrong with me or the baby. But this suits me and I really like the midwife who does the 3rd trimester ladies and deliveries here.
Also, if you are in Edinburgh, you only get one scan on the NHS, at 13 weeks, unless there is something wrong and even then, they will NOT tell you the baby's gender. You will need to pay privately for a 20 weeks scan, anyway.
Furthermore, again if you give birth at Edinburgh Royal Infirmary, you cannot book a private room.
Horses for courses.
Like Leonie, I prefer the British attitude towards pregnancy and delivery and could never deal with what I feel is the intrusiveness of the American system on the whole, but this is true of many things there and hence the reason why it's been over 7 years since I even went there at all and naturalised as a British citizen as soon as eligible.