You can use the axe by equipping it to one of the two item slots. Controls are posted on the inventory screen. One equipped, press the corresponding input (z or x) to use that equipped item. Also, only the smaller trees are choppable. Thanks for trying!
Yeah, I equipped it and z seemed to do nothing, if it is z on the keyboard.
If you equipped it to item 1 it would be Z yes - you gotta be rather close to the small trees to use it - there's no animation otherwise. I don't think i'll be spending more time on this, other projects are piling up - but lemme know on discord if you wanna see more :)