Hi,
I'd suggest you begin to look for a lawyer that's knolwedgeable for both US and UK legal aspects for this kind of situation. Complicating matters somewhat will be elements such as the ages of the kids and they may well also need specialist counselling in order to help them come to terms that you are both (currently) likely to split.
The reason I say this is that in my family circa 1997-9'ish we had a similar situation, a breakdown of a marriage, kids then aged 8 and 6, one US citizen and one UK/US dual nationality and one based in NYC and the other here. A good lawyer was found with experience of situations like this and 'generally' the split went 'alright'. There was an element of nastiness in terms of money, visitation, trying to wrangle the kids to one location or the other etc though.
I can't imagine how it all actually feels, I can only sympathize. If there's no chance of sorting things out, then find out exactly where you stand and all the potential aspects you ask above and more via the lawyers etc asap. At the very least, that will give you the entire picture specific to yours, hers and the kids so that you can make decisions as best as possible.
Good luck,
DtM, West London & Slough UK.