As title says, it appears that the AI does not reload them, they do pick them up off the ground but after they thrown 1-2 of them away they will not really do anything at all with them.
This occurred every time I had AI Bardin wield the throwing axes, I believe I tested it 5-6 times and it was present in 100% of those with the AI only getting <5 ranged kills.
In order to reproduce simply equip the axes and go into any map and let them throw the first 1-2 away.
The last two tries I took screenshots at the end screen showing their massive ranged kill count.
I know that having a overpowered AI that carries you through legend is not part of the plan in Vermintide, but equipping Bardin with the throwing axes should work as a valid ranged weapon in my opinion.