Prepare the white cake according to package directions in a 13×9-inch baking dish.
While the cake bakes, make the apple mixture. In a large skillet, whisk together brown sugar, water, cinnamon, nutmeg, and salt.
Heat over medium-high until bubbly, then add diced apples. Reduce heat to low and simmer for 12–15 minutes until apples are tender.
When the cake finishes baking, poke holes all over the hot cake using a fork, wooden spoon handle, or thermometer.
Pour the sweetened condensed milk evenly over the cake.
Spoon the apple mixture over the top and spread evenly, allowing the liquid to soak into the holes.
Refrigerate for 3–4 hours or overnight.
In a bowl, whisk pudding mix with milk and cinnamon until smooth. Refrigerate until ready to use.
When ready to serve, fold the pudding mixture into the whipped topping.
Spread over the chilled cake, slice, and serve.