17 seconds after bird impact, they used the "QRH Engine Dual Failure checklist" as per the NTSB report at:
Even though the engines did not experience a total loss of thrust, the
Engine Dual Failure checklist was the most applicable checklist contained in the US Airways QRH, which was developed in accordance with the Airbus QRH, to address the accident event because it was the only checklist that contained guidance to follow if an engine restart was not possible and if a forced landing or ditching was anticipated (starting from 3,000 feet). However, according to postaccident interviews and CVR data, the flight crew did not complete the Engine Dual Failure checklist, which had 3 parts and was 3 pages long. Although the flight crewmembers were able to complete most of part 1 of the checklist, they were not able to start parts 2 and 3 of the checklist because of the airplane’s low altitude and the limited time available.
The "procedure" that the captain followed was the mantra taught to student pilots very early in their training: aviate, navigate, communicate. The ultimate priority in all flight circumstances is to first fly the aircraft. By realizing that 1) continued flight was impossible, 2) attempting to restart the engines was futile, and 3) deviating to the softest and least congested improvised landing site was necessary, in fact in these circumstances the best possible option, the captain saved his own life and the lives of his crew and passengers.
The captain was almost a 20,000 hour pilot at the time of that accident and the first officer had accumulated over 15,000 hours of pilot time. This cannot be understated. This was an extremely experienced flight crew.
In many years of dealing with imminently life-or-death circumstances, I have seen two categories of people: those who act and those who freeze. Obviously, the crew in this situation was composed of people in the former category. That is the category in which I place myself, but thus far almost all of the life-or-death circumstances I have faced have been ones I have trained heavily to mitigate. My actions have been almost reflexive based upon that training, but in my craft I am also well over the 10,000 hour mark.
The ultimate answer to your underlying question of how did this flight end successfully is the right kind of experience of the flight crew, and this cannot be summarized in a checklist or procedure. They knew how to fly. They intimately knew the systems of their aircraft. Most importantly, they worked together as a team to accomplish the mission even though that mission had changed significantly since the time when they were cleared for takeoff.