I don't know if this helps you or not, but my husband is here on a work permit and I am a dependent spouse (from the US) and we were accepted into the NHS--no questions asked. They didn't ask to see passports, visas, work permit, National Insurance proof or anything. I called the surgery and told them our circumstances and they told us to come on down to register.
I was taking a long-term medication in the US and the doctor simply took my word for it and wrote a prescription for me here. (the medication happens to be somewhat cheap.) The one things they wouldn't do is give me my brand of birth control. If I wanted birth control they offered something besides Yasmin since it is expensive here...or I could pay out of pocket for it.
You may want to check about being reimbursed by your US insurer. I know my private cover in the US covers Out-of-Network, including in the UK. Should I need to see a private doctor for something that is not covered by my private medical here I know I have that as back-up. The downside is that I have to pay out of pocket and then fight with the insurance company to reimburse me for it.
Lastly, to see a private doctor here in the UK you must be referred by your NHS GP. I don't know what happens when you aren't eligible for NHS coverage.
If you have any questions i can surely try to answer them for you. If I don't know the answer, I can help you find out who can answer it.
Tracey