APU didn't save the day for Sully because at no stage Sully had dual engine flame out.
Hi vilas, The NTSB thinks it proved to be critical.
From
ntsb.gov/investigations/AccidentReports/Reports/AAR1003.pdf page 104,
" 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."
page 105,
"Although the flight crew was only able to complete about one-third of the Engine Dual Failure checklist, immediately after the bird strike, the captain did accomplish one critical item that the flight crew did not reach in the checklist: starting the APU.
Starting the APU early in the accident sequence proved to be critical because it improved the outcome of the ditching by ensuring that electrical power was available to the airplane. Further, if the captain had not started the APU, the airplane would not have remained in normal law mode. This critical step would not have been completed if the flight crew had simply followed the order of the items in the checklist."
Each engine generator would have dropped off line when N2<55% or when the engine master switch is set to off. The windmilling engines would still develop some hydraulic pressure. See Page 55.
"FDR data indicated that both thrust levers were set to the idle position at 1528:01, about 50 seconds after the bird encounter. The N1 and N2 speeds for the left engine both decreased while the speeds for the right engine did not respond. About 30 seconds later, the right engine master switch was moved to the OFF position.95 According to the Airbus FCOM, for an automatic start sequence, when the engine master switch is in the OFF position and the throttle is set at idle, the fuel valve will only open when the N2 speed is more than 15 percent when in flight. When the first officer attempted to move the right engine master switch to the ON position, the N2 speed was less than 15 percent. At 1529:27, the left engine master switch was moved to the OFF position, at which time the N2 speed was about 83 percent. The left engine master switch was moved to the ON position about 10 seconds later, at which time the N2 speed was about 39 percent."