The relationship and historical ties between the US and the UK cannot be accessed by individuals with immigration problems because the 'special relationship' is at the national level.
That means that any particular random person who comes along holding out America's hand for payment in order to ease their personal circumstances is not only doomed, but likely to be seen as laughably naive.
Now.
Having said that, I can put on my advocate's hat and point out that Americans are one of the largest ethnic groups in the UK. Not the largest, because that goes to the Asians, not the second largest either, but among the largest. At any given moment, there are an estimated 250,000 Americans inside the M25, be it naturalized citizens, WP holders, HSMP holders, students, spouses & dependents, and so on. That's nothing to scoff at, and moreover provides badly needed leverage for my own advocacy work.