Cosmo
.. (if) APU fully available at the time both engines ceased operations, would that possibly have had an impact on the situation? Specifically, provide instant Pneumatic resources to attempt a restart?
This is a time dependent judgement. If the engines have gone sub-idle and then the action might have been to turn off the fuel (crew shutdown) and start the restart procedure immediately following the ingestion. Whether you need an APU is subjective to how much altitude and airspeed can you maintain before you hit the ground.
However if the engines are still at or above idle it's a judgement call whether you try and work with a running engine or shut it down and start all over. With lots of altitude you can try several things out. At low altitude you quickly pick whatever is in your experience bank.
Maybe somebody at the NTSB hearing will ask Sully if he had an APU running or not would he have acted differently at the time (hindsight not allowed)
