I have found thy hostage lying on the floor and carried him outside but nothing has happened. He is in my arms and the quest still said to find him. I stopped to rescue other prisoners in a cage as I was leaving so I am wondering if this broke the quest?
Put him on a mount and being him much further away from the base. Normally if you ride far enough away from the location you need to rescue the person from, it will update the quest.
You will probably have to take him off the horse, throw him on the ground and he will stand up. Bit of a stupid system but it’s how it works.