I've only been pregnant in the UK so in terms of differences, I can only go by what friends and family in the US have told me about their experiences -- so, anecdotally:
The fewer scans thing is true but like Karin said, you can get private scans for not too much money. You may want to wait until later in the pregnancy in case you get sent for scans for other reasons -- I was thinking about having a private scan but now I'm having growth scans on the NHS every two weeks anyway (started 31 weeks).
I was surprised to find out I couldn't get amnio on the NHS unless they saw something disturbing at the 20 week scan, even though I'm 38, but maybe this is not an issue for you.
It did seem difficult for me to get any kind of prenatal care until my 12 week scan -- almost like no one wants to do anything for you until 12 weeks, even if you have lots of questions or some problems. But that was probably just my experience and if you have been pregnant before then you may not be bothered about this.
Once you know where you will be living, type your postcode into the NHS GP Finder and it will tell you your options for a GP (they don't assign one to you, you have some choice). Some areas will have lots of choice, others not so much. I think your choice for midwives is more limited but if you are really unhappy I think you can get them to switch you. In my case, my community midwives also work at the hospital, so I may have some continuity of care there.
In London you will probably have a choice of hospitals to give birth in, it helps to do some research. I'm going to St Thomas even though King's is a bit closer, for various reasons. St Thomas is very positive about VBAC from what I can tell, so that might be one to consider.
I do totally understand what you are saying because there are times when I'm really nervous about having a baby here. Like, they seem really fond of forceps here, which terrifies me... and my consultant OB isn't even a Doctor, just a Mister, which seems weird... and I've heard so many horror stories about the postnatal wards...
But you could totally have a bad experience in the States as well -- it's really just luck -- so I'm just trying not to worry about it. I think if you do some research and aren't afraid to be assertive when necessary, you can definitely have a good experience on the NHS.