Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or grease the cups.
In a large mixing bowl, combine the all-purpose flour, rolled oats, granulated sugar, brown sugar, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg. Stir well to combine.
In a separate medium bowl, whisk together the egg, buttermilk, vegetable oil, and vanilla extract until smooth.
Pour the wet ingredients into the dry ingredients and stir gently until just combined. Do not overmix; the batter should be slightly lumpy.
Fold in the peeled and diced apples evenly into the batter.
Divide the batter evenly among the 12 muffin cups, filling each about 3/4 full.
Bake for 22 to 25 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and allow them to cool in the tin for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.