Preheat the oven to 350°F (175°C).
Prepare a 9-inch round cake pan by greasing it lightly.
In a large bowl, cream together the butter and granulated sugar until light and fluffy.
Add the eggs one at a time, mixing well after each addition, followed by the vanilla extract.
In a separate bowl, whisk together the flour, baking powder, salt, ground cinnamon, and oats.
Gradually add the dry mixture to the wet mixture, mixing until just combined.
Fold in the chopped apples gently into the batter.
Pour the batter into the prepared pan and smooth the top with a spatula.
In a separate bowl, mix rolled oats, brown sugar, and chopped nuts for the crumble topping.
Spread the crumble mixture evenly over the cake batter.
Bake in the preheated oven for 45-50 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool in the pan for 10 minutes before transferring it to a wire rack.
Slice and serve the apple crumble cake warm or at room temperature, optionally with whipped cream or vanilla ice cream.
