Animations from Blender using default settings don't work in UE4 - help fix the Blender FBX exporter

The issue is not at UE. I say this because if it was, when you create a FBX using 3DS MAX or Maya for example, you would get issues, and this does not happen. The best approach for blender developers would be create something using Autodesk tools and export it, then the same job in Blender, and compare which are the differences and understand them. FBX format, thou proprietary, it is described by Autodesk, so the only way to achieve compatibility is with such comparison.