This is tricky because you are clearly on a budget way higher than average, and what you will want and enjoy will be very different to your average mumsnetter.
So, am assuming your DH is in private equity or some such and you lead a pretty privileged life, so my answer is based on that. But tell me a bit about what you are like and what you enjoy and I can help you pick an area which will fit.
SJW does have a longterm resident population, but unless you are very rich and Jewish you will struggle to connect with it. Your relocation agent is recommending it I suspect because of the American School there, which you are not yet ready for!!! It's not a young person's place either. Nor does it have good parks or anything.
So, either do Hampstead which is much more fun for young families, or stay close to your DHs work and go for the West London options. There are many many American families in NH and HP. Sort of depends whether you are left wing arty intelligentsia or not.
Really rich areas do have SAHMs because their husbands earn so much they don't need to work outside the home - though actually many of the ones I know do run businesses from home because they are the sort of women who had big careers before themselves and so want to do something themselves.
How long is the move for? How long is your rental contract for? Because maybe you should go somewhere fun and enjoy London and getting to know it for a short time, and then move after six months when you are hopefully about to pop. The very best way to make friends in a new area is to have a baby a you have probably guessed.