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. 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 the flour mixture and stir to combine. Add remaining flour mixture and stir. Stir nuts into the 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.