If Magus of the Moon loses its abilities, it continues to turn nonbasic lands into Mountains. This is because effects that change subtypes are applied before considering effects that remove abilities, regardless of the order in which those effects started.

Nonbasic lands lose any other land types and abilities they had. They gain the land type Mountain and gain the ability "{T}: Add {R}."

If a nonbasic land has an ability that triggers "when" it enters the battlefield, it will lose that ability before it can trigger.

If a nonbasic land has an ability that causes it to enter the battlefield tapped, it will lose that ability before it can apply. The same is also true of any other abilities that modify how a land enters the battlefield or apply "as" a land enters the battlefield, such as that of Vesuva or Cavern of Souls.

Magus of the Moon doesn't affect names or supertypes. It won't turn any land into a basic land or remove the legendary supertype from a legendary land, and the lands won't be named "Mountain."