WOAH... There are some mis-conceptions going on here...
First, the $80,000 foreign earned income credit for US taxes has nothing to do with the tax treaty. Anyone who does not reside in the US can deduct up to $80,000 per year from their worldwide earnings. The US will and forever tax US citizens on their worldwide income. Depending on your US permanent residence status you maybe taxed for 10 years after your departure from the US. In order to qualify for it, a person has to be a bona fide resident of a foreign country and "work" in the US for essentially a 12 month period. Once the 12 month period is met, the portion of the US tax year that applies can be deducted. Other deductions can also be applied, but everyone is different in that case. Generally speaking if you don't make over $80K a year, your federal tax burden will be nothing. You may still be liable for State taxes. Each state is different and you need to check this out.
The treaty comes into play in government retirement and pension schemes. Basically you can get credit for your US government retirement and visa versa with the UK.
The UK will only tax you on your UK remitted earnings. You must pay UK Inland Revenue taxes on any earnings remitted (brought into) the UK. What is remitted and what isn't can be a somewhat sticky issue, but for most people it would be what you earn while working in the UK.
Now those who are resident in the UK but not ordinarily resident and those who are temporarily resident (planning to spend less then 3 or less then 5 years in certain cases) have certain UK deductions available to them (like travel expenses to and from work) etc that can greatly reduce their UK tax burden. As to when you become resident in the UK, we recently discussed in the Taxes board here. I would recommend everyone take a look there.
But as for NHS... We have talked about this before in here as well. There are those who pay taxes, of which a portion goes to fund the NHS payments, and then there are those who are entitled to not have to pay for NHS care. Basically the public funds are paid to those who provide NHS services. Anyone who is legally resident in the UK for greater then six months is entitled to have their NHS services paid the moment they enter the country.