Adding to the very useful info the others have given, I would say that if you are American, you should definitely go into the private system rather than the NHS (whether you pay yourself or get insurance cover). If you are used to a high standard of care in the US, I think you have the potential to be very disappointed if you rely on the NHS system here. Obviously some NHS hospitals are better than others, but if you can afford it I would definitely say go private. I should add a disclaimer that I am pregnant with my first so haven't experienced either system first hand, so this is all based on talking to friends and reading online.
The Portland is run by an American company, so may be closest to the level of care you are used to in the States. By all accounts from friends who have given birth there, it is absolutely faultless. I was put off by the story circulating that there is no SCBU or NICU, which I subsequently found out to be false (I was going abroad just after I found out I was pregnant and had to make a snap decision, so didn't have time to research it independently).
Instead I opted for the Lindo Wing at St Mary's. Lots of the consultants practice at both the Portland and the Lindo, or the Lindo and the Kensington Wing, so the level of obstetric care is probably very similar across those three hospitals, and I would presume that they would be the three hospitals that you will shortlist (sounds like you live in Chelsea, so they are all within reach).
Lots of people rave about the Kensington Wing, but I had heard that the level of nursing care lets it down. This is just what I've heard on the grapevine and you may want to do your own googling/research into that.
Both the Portland and the Kensington Wing were refurbished a few years ago and the Lindo by comparison was pretty tired. However, the Lindo has just undergone a huge refurbishment and is about to fully reopen any day, so they should all be on level footing in this regard.
As Maples said, there are no birthing pools at the Lindo. Not sure about the other hospitals.
Also worth mentioning that the Portland appears to be considerably more expensive than the others.
As Ghislaine said, once you find out you're pregnant, you need to book a consultant asap, so probably best to make a shortlist in advance! By six weeks you probably won't get your first choice. When you book the consultant that automatically seems to give you a spot at the hospital (i.e. you don't need to separately book with the hospital).
If you want any more info or names of obstetricians that I've heard good things about, then feel free to PM me.