What nations own the moon?
No nation or nations own the Moon.
No nation owns the moon. The 1967 treaty named the Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space, including the Moon and Other Celestial Bodies, (also known as the Outer Space Treaty) explicitly forbids any government from claiming a celestial resource such as the Moon or a planet, claiming that they are the common heritage of man
None of them; by United Nations treaties, no nation owns the Moon, which is held as the common property of all mankind. Nice words, but meaningless. The settlers own the territory; they always have. If any nation were to have a claim to "own" the Moon, it would be the United States, since only American citizens have ever set foot on the Moon.
Several coalitions/partnerships of nations have landed or crashed robotic hardware on the surface of the moon. All 12 of the human individuals who landed and walked on the moon between 1969 and 1972 were citizens of the USA. Up to the present time in history, no nations have even temporarily left the Earth's surface.