Preheat oven to 350°F. Lightly grease a 9×13-inch baking pan.
Whisk cake mix, eggnog, eggs, oil, nutmeg, and cinnamon until smooth, about 2 minutes.
Spread batter in the pan and bake until the center springs back, about 28 minutes.
Cool the cake 10 minutes on a rack. Poke deep holes all over with the handle of a wooden spoon.
Whisk pudding mixes with cold eggnog for 2 minutes until slightly thick but pourable.
Pour pudding over the warm cake, nudging it into holes. Tap the pan to settle. Chill 30 minutes.
Stir whipped topping with vanilla, powdered sugar, and rum extract if using. Spread over cake.
Dust lightly with nutmeg. Refrigerate at least 1 hour before slicing.