Was the miracle on the Hudson saved exclusively by the APU? In the film Sully, (staring Tom Hanks) the APU is switched on straight after both engines rollback, why was this step necessary; Airbus said this was the "15th thing" to do on the checklist.

How or why did this step in particular prevent the loss of lives of over 100 people?
Yes, the solution to start APU was important.

The ditching procedure directs the use of maximum available slats and flaps for the final approach and touchdown (source, chapter 10.3). This is not possible without APU, as Airbus A320-214 cannot move flaps if only powered by RAT (only blue hydraulic line, same source, chapter 9.3). The running APU adds the green hydraulic line that can also move flaps.

Hence, without running APU, the optimal ditching procedure is not possible; that gives less chances for success.