Mix apple pie filling, 1 teaspoon cinnamon, and nutmeg in a bowl.
Beat cream cheese, confectioners' sugar, and 1/4 teaspoon cinnamon in a separate bowl until whipped and smooth. Stir apple pie filling mixture into cream cheese mixture.
Spoon apple-cream cheese mixture into each tortilla; roll tortilla around filling. Arrange rolled tortillas, seam-side down, in a 9x13-inch baking dish.
Mix white sugar, brown sugar, water, butter, cornstarch, and vanilla extract in a saucepan; bring to a boil. Reduce heat and simmer, stirring constantly, until sugar is dissolved, about 3 minutes. Spread sauce over rolled tortillas.
Bake in the preheated oven on the top rack until enchiladas are bubbling, about 50 minutes.