Quite a few work people have done small airport flights to the US, personally I'd always go down to Heathrow and leave from there rather than have to change flights, or have to use a small airline for a long haul flight - work colleagues used a Canadian budget airline to go to Vancouver from Manchester last year, and it sounded like a nightmare, 8 or so hours on the equivalent of Easyjet, without proper food, paying for drinks, cramped seats etc. Whatever you do, for the long haul bit of the flight, make sure it is on a big airline you've heard of.
Basically you have a choice of:
a)Hassle in the UK getting to Heathrow, then a single long flight.
b)Changing in Europe, meaning hassle in a foreign airport, plus two possible flight delays.
c)Changing in the US, again meaning hassle in a foreign airport when you are extremely jet lagged from a long flight, plus two possible flight delays.
Changing planes adds so much time that it is almost certainly worth getting train down / driving down to Heathrow time-wise. It also saves you from the depressing hell that is US airport departure lounges (I'm assuming you're not some fancy pants first class / frequent flier card type).
If you did do train down (with that selection of kids I'm guessing it wouldn't be easy!), you could do taxi from Kings Cross to Paddington and Heathrow Express. Or if you can brave the tube, Picadilly Line goes straight to Heathrow, is slightly slower, but cheaper and easier if you're going from Kings Cross.
I have an advantage in that I have a few relatives who live right near Heathrow - my preferred option is usually to stay over at someone's house the night before and then get a bus or taxi to Heathrow. I think splitting the journey that way makes the flight easier. I guess there are airport hotels if you don't have any friends down that way (possibly there might be good hotel & parking deals available if you look around).
Immigration at LA is a pain, in that you have to queue and be fingerprinted, but to be honest it is not that different from UK immigration - with a kid, the hard thing is the queueing, and you get that anywhere nowadays. They'll ask you stupid questions, but as long as you don't joke around with them, I've never had any hassle. Oh and make sure you fill in the landing card forms on the plane obviously.
I can use to sedate the children?
