So for everyone who thinks a wooden axe would work... have you ever -actually- cut down a tree? Like a real tree, of decent size. Not like a sapling. Something you had to think about when you were felling it. There is no way, absolutely no way, that you could use a wooden axe to cut down a tree. The edge simply wouldn't hold long enough to cut out a wedge. You're basically pitting the strongest part of a material against the weakest part of what is essentially the same material.
In theory, if you had a perfect piece of very hard wood (oak, yew, ash) that had been cured for at least a year, and shaped so that the fibers curved along the edge of the blade, and you were using this piece of master craftsmanship to cut down a recently dead and dry pine or spruce tree... Maybe. But you're not bringing down a healthy adult tree of decent size with a wooden axe. You might split a dry round into kindling with a wooden axe. But fell a tree? Hell no.
And honestly, a wooden training axe would be useless as a training weapon too. The weight would be completely different from an actual axe. It would be impossible to give a 100% wooden weapon the same sort of balance that an actual axe would have. Swords are one thing, since they are supposed to be balanced with the center of mass in the pommel. But an axe is going to have it's center of mass further up towards the striking end. Learning how to strike with a weapon that is balanced totally differently from the real thing is useless. I think that if wooden axes are present in the game, we should probably just assume that they have a striking head of some sort of stone, and that this is abstracted for gameplay purposes.