I've found a couple of sources that state the 727's APU cannot be started while flying, as it draws air from the wheel well.
It drew air from inside the wheel well, so it was only for use on the ground.
The APU on the Boeing 727 can be used for ground operation only.
Additionally, the battery on the 727 doesn't power all of the systems on the "essential bus", and presumably cannot start the engines.
So what happens if you lose all engine power in a 727? How can the engines be restarted?