A full transcript hasn't been released. The NTSB did give a summary:
The crew responded: "Ah, Cactus 1549 … hit birds, we lost thrust in both engines. We're turning back toward LaGuardia." At 3:28:05, 33 seconds later, ATC asked if the crew wanted to return to LaGuardia. "We're unable. We may end up in the Hudson," came the reply, according to Higgins' reading of the transcript.
"There was some discussion whether the airplane could land at Teterboro, off the airplane's right, about six miles. And the pilot responded, "We can't do it … we're gonna be in the Hudson.'" That was the last communication from the aircraft.
They also said that the word "Mayday" was on the CVR but not on the ATC recording -- they may not have keyed the mike.
The captain took over after the strike and flew and operated the radio while the first officer tried to complete the restart checklist.
Even if a performance study shows they could have made it to Teterboro, the captain had to a make a split second decision with a high penalty for failure, so I don't think he can be blamed for choosing the river instead.