London, indeed England is a wonderful place to live OP . London is probably the closest thing to NYC, outside the US, anywhere in the world - so you'll be fine. People are very friendly, humorous, convivial and neighbourly on the whole. You could be years discovering London, it's places of historic interest etc. It's huge, ancient and jam packed with history and character.
I came here as a 6 year old from rural Italy and apart from a year in Rome as a student, I wouldn't/couldn't live anywhere else.
I understand entirely it must feel daunting for you, moving all the way across the pond. But if I may offer my thoughts coming from my own experience? London is an incredibly multi-cultural city and you will find a community of every nationality here – from Portuguese to Korean. Nigerian to Peruvian. As you know, there is a considerable American community too.
My ex partner of many years and the father of my 2 daughters, is French. He has lived here 30 years and all that time, purposely kept a respectful distance from the French community here in London (possibly the largest of all non – British born communities). Indeed we had French friends who had lived here for many years, for whom I was the only English person they knew. Their children went to the French Lycée, did sports in French clubs. Socialised only with their fellow French, bought their school books and groceries from the exorbitantly priced French retail outlets in South Kensington and went to France for every school holiday. This meant that London and Londoners for them always remained a somewhat foreign place and species, that they never felt a part of, or really got to know. Their children, in the main, didn't have any English friends, rarely even encountered English people and didn't feel particularly at home here either, despite some of them having been born here. London went completely undiscovered for them. Very rarely did they venture anywhere in London let alone the rest of the UK (which is wonderful too), or beyond their French 'bubble' of friends and South Kensington.
This would be a great shame and a missed opportunity for any family in my opinion. To grow up in London always feeling somewhat an outsider. London being 'out there', so to speak, at arms length and something that goes on without your participation. I'm not suggesting for one moment that this is how you intend to live, but I think it might be a bit of a danger!
It goes without saying that your children's experience of being here will differ from your own, so my advice would be, if you are able, to live in an 'average' London area, where you will doubtlessly bump into Americans anyway. But somewhere where your children can mix with English children, (whom I'm sure will find them fascinating being American) make lasting friendships with them, so they can keep in touch with, visit them and have contacts here in later life. I think it would be life enhancing and mind broadening for them and you, to really immerse in this country whilst you are here. Plus yours will develop English accents which I believe is quite desirable in the USA? 
Anyway Ive said far too much, I'm sure this has all been covered and I'm not telling you anything you don't already know .. So here's wishing you all the very best anyway... Best of luck, you'll love it here!!