Preheat the oven to 350°F. Join the butter and the sugar in an electric mixer and cream well until fluffy. Add the egg and beat to integrate.
In a large bowl, mix up together the flour, the baking soda, the cinnamon and the nutmeg. Then, with the mixer on low, slowly add the dry ingredients to the butter mixture until incorporated (do not over beat).
Remove the bowl from the mixer and stir in the apples and nuts. Join all together with a spoon.
Pour the batter into a buttered 9-inch cake pan and bake for 30 to 40 minutes, or until browned and a toothpick inserted into the middle comes out clean. Remove to a rack and cool before cutting.