Combine flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Stir in nuts.
Cream butter and sugar together. Stir in eggs, one at a time, mixing well. Stir in vanilla.
Peel, core, and chop apples. You should have about 3 cups.
Stir dry mixture and milk alternately into creamed mixture, starting and ending with dry mixture, making 3 additions of dry and 2 of milk.
Fold in the chopped apples.
Bake in the greased and floured 13x9-inch baking pan in a preheated 350°F oven for about 35 minutes, or until a tester poked in the centre comes out clean. Cool 10 minutes in the pan before cutting. Cut into 32 pieces.