I moved over in February, just before the requirement to have insurance, and when I spoke with someone in our states health insurance group about me not having insurance, they said it will be fine. The fact that you will no longer be living in the US makes it to where you are not subject to the health insurance requirement. Or so I was told.
You are eligible for NHS on your Fiance visa, I used NHS in the second month I was here. When registering with a GP you need your passport/visa and a bill with your address on it, so get your name on a bill at your address as quick as you can. You'll need that bill for a lot of different things like opening/being added to bank accounts, application for your NI Number (needed for work), Registration of Marriage, etc.
My local surgery was nice enough to get me to where I could see them without one, I just didn't get my NHS number/card for about 5 months. If you can be added to the water bill, that would probably be fastest. Many of the electrical companies only allow one person per bill. And local councils aren't always great about getting you on the bill right away. haha. At least mine wasn't. *wry grin*