I'm with Capt Claret ... the most prominent issue for me was why the Company was asking such a wankish ( I'm sure it's a word) question of the crew at such an innapropriate time
From my one listening of the tape, I think ATC were advised a lot earlier about the problem than is obvious. You can tell by some of their questions that they know the story. I think a fair bit was not included. In fact, I don't think the phrase engine failure is ever mentioned ... so obviously it was discussed earlier.
P.S. I just listened to the 2nd recording ( the video) and they advise ATC straight off that there's been an engine failure. That was not in the first audio recording.