I think that moving with very small children to a place that you don't know, where you have no connections, would be incredibly hard in most circumstances. The only way that sort of move works is where you have a totally solid relationship and you both are really up for it. As you have an unsupportive husband (even if he was not a bully) I think that you would be really unwise to make the move.
dh and I moved to New York last year, and then on to Canada when that didn't work out, and I really don't think that most of North America, barring some parts of some cities (New York, San Fransisco, Chicago maybe) would give you anything like your London life. That's not to say that there is noting to offer here, but it isn't at all what you'd get in London.
Put it this way, my sister has just moved from Stoke Newington to Melbourne, a city which many people think is fabulous. But she really really misses London, where she had friends in walking distance, a job she enjoyed, knew everything that was going on, where to go etc (prob very similar to the OP) lots of similarly minded people around, and probably where would have been happy to live the rest of her life. She is the most positive minded person I know, but it's clearly a huge wrench (and as her dh is from Melbourne they have connected to friends and family).
My view as an "ex-pat" is t hat you have to have a really strong "pull"to where you are going, whether that's because it offers something that your current location doesn't, or because you are up for an adventure. Otherwise most likely you will not be happy.
To some extent I think that the OP needs to ask herself whether she is going to say good bye to her marriage now, or sometime next year, or whether she is prepared to be totally miserable to keep it going. Because you never know Nashville might be really fun, but chances are that it might be a very miserable experience.