Preheat the oven to 400 degrees F (200 degrees C). Line a muffin tin with paper liners.

Combine oats, whole wheat flour, all-purpose flour, white sugar, baking powder, baking soda, salt, cinnamon, and nutmeg in a medium-sized bowl; set aside.

Lightly beat eggs in a large bowl. Stir in milk, oil, brown sugar, and vanilla extract. Add bananas and mix together well.

Add 1/2 of the flour mixture to egg mixture; stir to combine. Add remaining flour mixture and stir. Stir nuts into batter.

Spoon batter into the prepared muffin cups, filling each 3/4 full.

Bake in the preheated oven until a toothpick inserted in the center of a muffin comes out clean, 18 to 20 minutes.