How does Aurane find and train thousands of sailors and crew in only a few months, when there is no way he could with prisoners alone? Fellow survivors of the Blackwater? Redwyne recruits? Commercial seamen? Randoms from the city? Men from Driftmark? Some combination of the aforementioned? Some other way?

I think Aurane largely recruited from the ranks of trained sailors, rather than taking the time to train landlubbers. Keep in mind that King’s Landing is both a royal naval base (lots of sailors whose ships went up on the Blackwater who need work), and a commercial harbor (with a lot of the piers and wharfs by the Mud Gate destroyed, those civilian sailors are also looking for work).