Rob, I think you have two different issues to solve.
First, your UK company. I am fairly sure you can run it from anywhere in the world, less sure on the fiscal consequences. The UK tax authority might not be happy with it, revoke tax benefits, and the like. Check with local tax advisor to find out.
Second, your immigration status in the US. I was able to get a B visa, which is a visa for people who have to stay in the US for an extended period for personal or mixed reasons, do not work or look for work in the US, and are planning to eventually return to their home country. I think you have good chances to qualify for this visa since you can present your company as evidence that you have an interest in going back home, and that you can financially take care of yourself. The downside of the B visa is that you are allowed to stay only for periods of 6 months. Meaning that after six months you have to exit the country. You can come back with the same visa, but if you play this game more than twice immigration officers might become suspicious of "immigration intent" and have the right to deny you entry in the US, notwithstanding the visa. It never happened to me, but for sure they asked me a lot of questions at the border. As per life in the US, you won't have a SSN (social security number) so you would not be able to get a US credit card, but you can open a checking account and get a debit card, checks, etc.
In any case, check with an immigration lawyer (or just the US embassy in London) what would be the best strategy/visa for you.