Preheat the oven to 325°F. Grease a 9-inch springform pan.
In a medium bowl, combine graham cracker crumbs, melted butter, and 1/4 cup sugar. Press the mixture firmly into the bottom of the prepared pan to form the crust. Bake for 10 minutes, then remove from the oven and let cool.
In a large mixing bowl, beat the softened cream cheese with 1 cup sugar and vanilla extract until smooth and creamy. Add the eggs one at a time, mixing well after each addition. Pour the cream cheese mixture over the cooled crust.
In another bowl, combine the diced apples, cinnamon, nutmeg, and brown sugar. Spoon the apple mixture evenly over the cream cheese layer.
In a separate bowl, mix together the oats, flour, and chopped nuts (if using). Sprinkle this crumble mixture over the apples.
Bake the cheesecake for 50-60 minutes, or until the center is set and the edges are lightly browned. Turn off the oven and leave the cheesecake inside for an additional hour to cool gradually.
Remove from the oven and let it cool to room temperature, then refrigerate for at least 4 hours or overnight before serving.