I'm over on the other side of South London & only know Fulham from trekking across to English National Ballet School & Dirty Dance Attic for classes. My friends who live/have lived over that way all love it though.
Not that I'm a TOTAL bunhead or anything, but if your kids are into dance (or want to get into dance!) I very much suggest you contact the West London School of Dance (www.westlondonschoolofdance.co.uk). The school's Artistic Director has the, er, pleasure, of being my main teacher these days & she takes a very real interest in the children from the baby ballet class up to the children graduating from the linked vocational school. Those of her teaching staff I know are incredibly passionate as well as of course being well-qualified, experienced & gifted teachers & I'm certain those I don't know are the same. (And no I'm not on commission. I'd suggest it, but I don't fancy a round of extra sit-ups for such outrageous cheek...). It seems as if the parents of the students build some pretty strong & lasting friendships, too...
Thanks to sieve!brain I may be totally imagining you said your children are 6, 4 & 2? But didn't give their gender? The oldest would be old enough to be involved in either Scouting or Girlguiding (if a girl, totally send her to Rainbows because clearly vastly superior to Beavers & Girlguiding generally awesomer than Scouting & if you were able to volunteer to help you would definitely make some new friends
. (Actually, are you QUITE sure you don't fancy SE London instead? Blackheath is LOVELY. Plenty of good schools & good public transport links. And a very nice Brownie Unit up the road where you would be loved FOREVER if you came to help. No? Well, can't blame an owl for trying...)
It must all seem quite daunting, but It Will Be Grand. Please don't let your friend's experience put you off - try not to carry it in the back of your mind as you'll find it niggling at you & pushing you to overanalyse every interaction with people & have you worrying a week in that you're doomed to friendlessness for your whole stay because you've not found anyone you've clicked with yet.
You've had heaps of fab suggestions on here about possible places to go - I'd suggest getting yourself a notebook (not just because I'm a stationery fiend, I promise...) so you can make notes of the various suggestions & look up the relevant contact details & if you get in touch before you move over make notes of conversations/meetings arranged etc. You might feel a bit more confident if you had some plans in place? Certainly knowing there are some people to make contact with when you move over might help?
It will, as other posters have said, be a bit of a culture shock - two cultures divided by a common language AND SO MUCH MORE... Try not to let on how shocked you are though as it does tend to be interpreted as criticism & upsets people. Like the love of all that is good & holy do not suggest anything along the lines of your having expected the NHS to be "like something from a third-world country" because, as a PP has said, we are (rightly) proud of it & whilst we might grumble amongst ourselves, criticism from outsiders is intolerable. (I got Really Quite Cross with somebody on Facebook only this afternoon for their [unwarranted & Very Rude] attack on our glorious NHS, in fact.)
The British are, generally speaking, more reserved than our counterparts across the pond. We're not being unfriendly/cold, we're just less... expressive?... about things. Quieter, almost? But that's not exactly what I mean, because it's too close to the idea of the Brash Loud American, which ISN'T what I'm getting at. We're just not quite so free & casual with people we don't know, I suppose? Which is, actually, mostly to do with being polite & respectful. There are, of course, exceptions to this, just as there are doubtless plenty of Americans who are more reticent!
It's worth Learning The Language before you move & making it a game for your kids. One of my friends made the reverse move (as it were) when she was a little older, my cousins lived in the States for a year when we were very wee & I've got friends who worked in summer camps as older teens. All of them found it either REALLY helped or, unfortunately, really WOULD have helped, to Know The Language. It will depend on how many other American children are in their class, but if they refer to their trousers as pants, they are likely to be laughed at. Hard. For some time. It helps if they don't hesitate at "rubbish in the bin" rather than "trash in the trashcan" & if they won't be confused it's T for Torch not F for Flashlight. I don't know if you ever use pavement to mean the surface cars run on, as they do in Canada, but here it is what pedestrians walk on - my lovely Canadian Guide Leader's son was VERY confused about being told to walk on the pavement instead of the sidewalk by his teacher early on in their year in the UK, but he obediently did so & was then even more confused at being told off...
You'll want to get Oyster Cards sorted for you & your DH as soon as possible - before you move over here if that's possible (his work should be able to advise). They are the way public transport is paid for in London - you can no longer buy paper tickets on buses at all, just to warn you. Excitingly though (well, I'm excited, it means we can now more easily run all sorts of Brownie outings FOR FREE!) children aged under 10 no longer have to pay to travel on the national rail network where pay-as-you-go Oyster card payments can be made. They could already travel free on the tube, bus, tram, overground & DLR (Docklands Light Railway, a driverless train that serves East London) so in Fulham you'd have been fairly sorted for most things, but this opens up all sorts of possibilities for Adventures...)
Sorry, I've rather gone veering wildly off the point. But I'm sure that you're going to be Just Grand. You sound from your posts like you'll fit in just fine. And London is full of people of different nationalities, some here to stay & others just passing through. If you ever feel like you're standing out like a sore thumb then I apologise in advance for it because that'll be the work of some twitface making you feel uncomfortable - well, unless possibly you're prancing along the Mall wearing nothing but some reindeer antlers singing a medley of Broadway Songs. That one would be on you...
Good luck with it all & feel free to PM me if you'd like to know about London life from the perspective of a lifelong Londoner (barring working in Lancashire for some of my year out & then being away at uni). I can't help with the expatty stuff, but London Life I should be able to help you navigate if you'd like. I promise (Guide's Honour) I'm not a weirdy one. Just a horribly overtired & a bit daft with it one...