You haven't described it too fully as to size, but if it is about 7.92mm diameter, it is probably for a German rifle and machine gun blank cartridge.
The "running out of materials at the end of the war" is a common myth. The wooden bullet disintegrates immediately on leaving the muzzle of the weapon and was not intended to be lethal.
If it is about 20mm diameter, it is for a blank cartridge for the German Flak 38 20mm gun.
If a soldier was hit by wooden fragments, they probably came from some wood near him that was hit by something else.
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