(Eek. Sorry for what has turned into a long post, but reading the thread has made me a bit worried about you ... I hope this gives some food for thought and doesn't come across as a "you're a terrible mother" post like some of the above perhaps have.)
It's just ... Even if you forget the money (let's assume you win the lottery) it is still a HUGE thing to plan based on a relationship that isn't even a few years old yet. I'm over a decade in to a relationship with my DH - we were in love right away, have barely spent a day apart since then, and I'm still learning who he is under pressure and more.
You don't really know this guy yet, and yes - time together does count. You can't fast-forward a relationship. Some people could marry after a week and watch it last, but I would bet everything I own that most would regret it.
On a practical note, five years from now your kids will have school friends and be very settled in their lives here. Moving to a whole other country to live with a new man as "dad" who they've never lived with before is a massive life change and will cause problems, it just will. I say this as someone messed up partly because of moves across schools within the UK in one steady family. Even as adults, DH and I moved 2 hours away and felt so lonely and lost that we eventually moved back. There are so many schooling and moving issues in your ideal plan that I don't know if you've thought about - and kids are adaptable but there are limits and so many variables that make a difference.
Having said "forget the money", realistically if you don't have a good job lined up and can't sustain yourself without the man's help you could be totally screwed. What happens if he's knocked down by a car? Healthcare is huge too. Everyone's healthy until they're not and two lost, frightened homesick kids may well develop more bugs and stress-related illnesses than you've ever noticed before.
Honestly (and I don't mean this to sound mean because it sounds like you're still getting over an abusive relationship and being a single parent which must be so tough) ... this really really sounds like daydreaming and imagining yourself out of your current life into a perfect idealised one, with a "safe" and idealised bloke who you can (for now at least) keep at arms length in a long-distance relationship.
The problem with daydreaming is that it's a distraction and isn't dealing with real issues. It's also a fuck of a slap in the face when it ends. People are asking about your relationship because you want to move to the USA to escape, but there is no escape - any issues you have won't change just because you're living on a different bit of rock - in fact they're more likely to be exacerbated.
I'm more than a bit worried about your partner too if he really thinks this is all simple, viable, and has absolutely none of these concerns. You just seem so, so vulnerable - please keep thinking things over.
And remember, you're not the only one who could get caught up in this daydream. If you spend 5 years telling your kids that life here is all temporary ("wait till we move to America!") and then you don't (maybe money, maybe a breakup, maybe something else) - well that will cause issues too. I still remember when we were finally settled somewhere, we had a long year of "we're moving to X" - visiting a new school, getting ready to say goodbye to friends, packing, and then we stayed put. It sounds like such a little thing but it can make an impact when you're young.
anyway - it sounds like you've had an absolutely terrible time of it and I'm not surprised you've projected some feelings on to the whole of the U.K. as a result. And congrats if you made it to the end of the longest thing I've ever posted to MN!