I made this small concept without using anything custom stuff just with blueprints.
- Trees PCG Static meshes with a material function implemented.
- A CutActor, that detect static mesh overlaps, controls overlapping trees material function. In background creates a dynamic mesh as filler copied to StaticMesh. Spawns some splinter vfx on vertices.
- CutActor on cut creates in the background, top and bottom boolean meshes and ready them to be spawned on cut.
- (Didn’t push futher) On cut command we can push step the cut object. After a treshold is reached (bounds volume or cut interaction number). Top and Bottom meshes are made visible and phsyics would be applied to top. A problem over here to access pcg spawn to be hidden or removed.
- Sound of cracking, swoosh tree falls, physics disabled, player gets loot, tree is removed with material.
If a company pays me I do it 
but can share material functions and raw protoscripts for cutter (which is ugly) however these whole concept can be done also just using dynamic meshes. It is possible to detect on cut angle have a collision and detect if cut is fully reached. Howerver in terms of fun maybe it is too much effort for player decreasing its fun, On the other hand there is an opportunity there in terms of item crafting/blacksmith to have bigger better axes lowering effort spend pushing player into getting stronger with craftsmiths.