Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or grease the cups lightly.
In a large bowl, combine the wheat bran, all-purpose flour, brown sugar, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg. Stir until well mixed.
In a separate medium bowl, whisk together the buttermilk, vegetable oil, egg, 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 finely chopped apple pieces evenly into the batter.
Divide the batter evenly among the 12 muffin cups, filling each about 3/4 full.
Bake in the preheated oven for 18 to 20 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and let them cool in the pan for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.