Preheat the oven to 350°F (175°C). Grease an 8-inch round cake pan and line the bottom with parchment paper.
In a medium bowl, whisk together the gluten-free flour, baking powder, baking soda, cinnamon, nutmeg, and salt until well combined.
In a large bowl, beat the granulated sugar, brown sugar, applesauce, and vegetable oil until smooth.
Add the eggs one at a time to the wet ingredients, beating well after each addition.
Stir in the vanilla extract.
Gradually add the dry ingredients to the wet ingredients, alternating with the dairy-free milk, beginning and ending with the dry ingredients. Mix until just combined.
Fold in the diced apples gently.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before serving.