Are cinnamon and nutmeg interchangeable in desserts like apple pie, apple crumble and rice pudding?

No. They taste very difference, but they go together.

 Cinnamon is warm, woody, hot, sweet, spicy, bark. Nutmeg tastes like eggnog. By volume, you can use much more cinnamon than nutmeg. Nutmeg is "a dash of" kind of spice, while cinnamon can be mixed with butter or sugar and slathered on basically any pastry.

The tastes may be different but YES

If a recipe says Cinnamon, and you only have Nutmeg (or you just prefer the taste of Nutmeg) then you could use that instead .. and vice versa.

Almost any Herb or Spice is 'interchangable' in a recipe .. depending on what you have (or prefer).

They have different flavors, and the flavor of nutmeg is much stronger than cinnamon, so you use much less nutmeg than you would cinnamon.

No, I would not use them interchangeably.

For those desserts, yes. But not for most everything else. 

To my mind they are interchangeable in those sorts of recipes, however individuals may have a strong preference in either direction.  For example, I like rice pudding both ways favouring nutmeg but my mother wouldn't touch it at all with nutmeg.  She'll wince if she sees me adding any to my portion.