The essence of the American special relationship is that America is special. This should not be any surprise to anyone who thinks about what it means to be a king. The president of America is a king in the real sense of wielding the power and authority of the most powerful state the world has ever known. His sworn responsibility is to maintain that power and dominance. We have a monarch in name, but in reality the king of the united kingdom is also an elected official, the prime minister. In some ways he has less personal power or freedom of action than the president, but in other ways he has more.
No one should ever imagine that America will do the slightest thing against its own percieved best interest. While the great threat to the world was russia, this made most of the rest of us allies. Now Russia is no longer Americas great threat. Many smaller issues worry the Americans. One of them is terrorists, another is guaranteed oil supplies. A third is the growing power of the EU which sometimes refuses to go along with what America wants.
Reagan and Thatcher were allies. They had similar political outlook and shared a common enemy. Most things were right for them to get along well and fight the same cause. Bush and Blair have rather different motives.
In Hitlers case, I think he enjoyed a happy coincidence of conviction AND personal gain.