Preheat oven to 400°F. Grease ten 3 x 1 1/4 inch (3 1/2 to 4 ounce) muffin cups. If you have empty cups in your muffin pan, put 1/4 to 1/3 cup water in each of those cups.
In a medium bowl, stir together first 5 ingredients, until blended. Stir in bran morsels, and let stand 1-2 minutes, or until cereal is softened.
Meanwhile, in another bowl, stir together next 5 ingredients. Make a well in center of dry ingredients; add bran mixture, and stir, just to combine. Stir in apples.
Spoon batter into prepared muffin cups; bake 15-20 minutes, or until a toothpicks test clean in center of muffins. Remove tin from oven to wire rack. Cool 5 minutes, before removing muffins from cups; finish cooling on rack. Serve warm or cool completely, and store in an airtight container at room temperature. Makes 10 muffins.