In 1956 Britain was the key player in the Mid East not the US. One background reason for Suez was because Nasser opposed the British led Baghdad Pact, he didn't want the Mid East centre of gravity moving from Cairo to Baghdad. If the both sides dug in their heels Britain and France could default on their loans and go back to being independent powers, and then where would the US be? There is nothing set in stone about economic dependence on the US, it was done because it suited all parties and was easy.
You don't get it, not only would they be pissing off the biggest economic power in the world, they would also be pissing off everybody else too, even the Canadians were pissed, Saudi Arabia cut off oil to both countries, and nobody wanted to make up the difference by increasing oil shipments, including other NATO members. This is aside from the fact that defaulting on federal debt would effectively ruin their ability to deal with other countries financially, and if the US pulled backing of the pound it would effectively destroy british domestic policy. The welfare state would have been aborted in the womb, long before the thatcherites came into being. It would be political suicide for the Eden government at the very least.