If you are a dual UK/US citizen and are returning to live permanently in the UK, you will be entitled to NHS care from day one and do not need health insurance.
You should be able to use emergency or walk-in services immediately, but in order to fully register with an NHS GP, you'll need proof of UK address... however, you should still be able to get treatment as a temporary patient if you need it before you can provide proof of address.
It won't take a few months to get proof of residence though... you just call the local council or utility companies, get your name added to the accounts and then wait until you receive a letter in the mail with your name and address on it... shouldn't take more than 1-2 weeks, maybe a month or so.
Considering that the NHS receptionists are not used to charging anyone anything and usually have to be hounded to give you a bill, it's unlikely they will try to demand payment.... they are more likely to insist that you don't have to pay, even if you know you do have to pay (but if you are a UK citizen and have moved permanently to the UK, you won't need to pay anyway).