I had baby #1 under the NHS in London and then, as already said, #2 in the US.
The obstetric care under the NHS was wonderful during labor/delivery: cannot fault it.
Having had private care in the UK for non-obstetric issues, I naively expected medical care in the US to be similar. It really wasn't. I could easily have been giving birth in an NHS hospital in terms of things like decor, food, quality of bed sheets, wait for scans (not during labor but H had a 3hr wait for my 20wk scan for example...). The only difference was that it was my OB (whom I'd worked with throughout my pregnancy) who delivered my baby: none of the NHS randomness.
Medically I don't think there was any difference whatsoever.
My point is that if you're going private for a 'better' experience, then you may well be very disappointed.
And aren't you allowed to elect which NHS hospital you want to give birth at? So the OP could choose a different hospital to the one she gave birth at first time around?
Also factor in wait time for a passport. US hospitals submit the birth certificate application whilst you're still there after the birth, and we then paid for an expedited passport service, but it still took around a month. You could be hanging out in the US for a long time.
Another thought: who's going to pay for your 6wk check up (you're likely to still be in the country then)? For the regular pediatrician appointments for the baby? What happens if, for example, your stitches don't heal and you need to be seen? Or you develop PND? Not nice to think about, but how will you get (and pay for) treatment?
The more I think about it, the more I think it's a silly idea to be honest unless you have unlimited funds...