All my osage is salvage osage, cut in front of or behind a bulldozer. My osage cutting days are over, I have two lifetimes stockpiled, most of which I will probably give away as my age advances and I just can't make bows anymore, unfortunately I am almost there.
I have cut osage year round out of necessity, cut it or lose the opportunity. Like the others I prefer colder weather but don't remember having that option very often.
You will hear "cut it in the winter when the sap is down", nonsense, it doesn't matter, the osage I cut in N Alabama runs sap year round. You will also hear" it will burn your chainsaw up it is so hard" more nonsense, green it cuts as easily as any other hardwood. If you see sparks fly from a green tree as you cut it it has grit in the bark. A good sharp blade is a must as with any wood cutting, Seasoned osage is a different animal and pretty hard stuff, but again, a sharp chainsaw will do it in.