Preheat oven to 350 degrees F. Generously butter and flour bundt cake pan. Tap out any excess flour. Set aside.
In a large bowl, whisk together flour, baking powder, baking soda, nutmeg, cinnamon, and salt. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment (or use a handheld mixer), cream together butter and sugar. Scrape down bowl as needed. Add eggs one a time, ensuring each egg is incorporated before adding the next. Add rum and vanilla. Mix until combined.
Add dry flour mixture in three additions, alternating with the eggnog. Begin and end with the flour mixture. Mix on low speed until thoroughly combined. Scrape down bowl as needed to ensure thorough mixing.
Pour batter into prepared bundt cake pan. Level off batter. Bake for 55-65 minutes until cake springs back to touch and toothpick inserted cake comes out clean.
Allow cake to cool in pan for 10 minutes. Invert bundt cake onto a wire rack. Remove cake from mold and allow to cool.
To make rum glaze: whisk together powdered sugar, cinnamon, and nutmeg. Add vanilla, eggnog, and rum. Whisk until smooth. Pour over cooled bundt cake.