Preheat the oven to 400 degrees F (200 degrees C). Line 12 muffin cups with paper or foil liners, or grease if not using liners.
Combine oats, all-purpose flour, whole wheat flour, baking powder, baking soda, cinnamon, nutmeg, salt, applesauce, egg whites, milk, brown sugar, oil, and vanilla in a large bowl. Mix until blended. Fold in raspberries, blueberries, and blackberries; batter will be lumpy and thick. Divide batter evenly into the prepared muffin cups.
Bake in the preheated oven until a toothpick inserted into the center of a muffin comes out clean, 15 to 17 minutes.
Let muffins cool for 5 minutes in the pan before transferring to a cooling rack.
