No matter what, I would move the shed. The current governor of Texas was paralyzed when he was hit by a falling tree. If you are in the shed, you will get little warning. You may think that the tree will only fall over in a storm, but I have seen trees fall over on a clear, calm summer day on my own property. Big trees, about the size of that one.
If you choose to keep the shed where it is, your shed and contents should be covered by your insurance. Be sure it is sufficient, make certain you have good records of what is in the shed (pictures are very good). If the neighbor won't cooperate, let the insurance companies fight it out after something happens.