Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or grease the cups.
In a medium bowl, combine the wheat bran cereal and buttermilk. Let sit for 10 minutes to soften.
In a large bowl, whisk together the whole wheat flour, all-purpose flour, brown sugar, baking soda, baking powder, salt, ground cinnamon, and ground nutmeg.
In a separate small bowl, beat the egg with the vegetable oil and vanilla extract.
Add the egg mixture to the soaked bran and buttermilk mixture and stir to combine.
Pour the wet ingredients into the dry ingredients and gently fold together until just combined. Do not overmix.
Fold in the 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 allow them to cool in the pan for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.