Just went through this for the first time today, but I got it to work after the quest failed only once. Obviously, your mileage may vary, but this is what worked for me:
- Got quest in Azshara, boarded zeppelin & waited for it to depart and trigger the travel load screen.
- As soon as the game returned from load & I was on a now-empty zeppelin in Twilight Highlands, I logged out. Character sat on the deck of the ship & the 20 second countdown ticked off normally. (Note: I chose 'Logout', not 'Exit Game')
- As soon as I got to the character select screen, I immediately hit the 'Enter World' button.
- Loaded back to the zeppelin, which was now full of NPCs and surrounded by the air fleet.
Everything worked as intended from that point on, and I was able to parachute down to the ship. Like I said, your mileage will surely vary, but hopefully this will help save someone else from an eternity of retries.
One other (possibly important) note: The first time the quest failed, I rode the zeppelin ALL the way through to the mission end location without anything happening. I kept waiting for something to work but... nothing ever did. But at the end, all it did was pause for a few seconds & then reload back in Azshara, flying back to the tower. So I just stayed on the zeppelin & rode it back around a second time. I never abandoned & reacquired the quest or anything like that.