All they have to do is move outside the nation's territorial boundaries.
And no nation “owes” Outer Space.
All they have to do go out and physically stake a claim and start mining operations it, not simply say they own it as that fellow in the article ridiculously did.
Of course, getting the material in nations that prohibit it may be a different matter...
How can something that is not of Earth be the communal property of all Earthlings?
That is insane. If no person (Earthling or non-Earthling) has set foot or occupied this planetoid how can any individual or communal body lay claim to that which they can not physically possess.
The only law that I can see having any relevance to this argument is the law of salvage.
I would consider an asteroid analogous to a derelict ship at sea. Anyone finding the asteroid and take possession of it can claim salvage rights to the asteroid.