Preheat oven to 325 degrees. Generously spray the inside of a 12 cup bundt pan with baking spray, set aside.
In a large bowl, combine flour, sugar, pudding mix, baking powder, salt, and butter. Beat on low until mix resembled small pebbles.
Add the eggs, milk, rum, oil and vanilla extract. Beat until well combined and smooth, scraping the sides as needed.
Evenly sprinkle the chopped pecans/walnuts into the bundt pan. Pour in the cake batter, tap on the counter to remove air bubbles, and bake for 55 to 60 minutes until golden brown and toothpick comes out clean when inserted into the center of the cake.
Remove cake from oven and set on a cooling rack while you make the Rum Syrup.
To Make the Rum Syrup
In a small saucepan, combine the butter, water, and sugar.
Bring to a boil, reduce heat and simmer until sugar is dissolved and begins to thicken.
Remove from heat and stir in the rum.
Slowly begin to pour the syrup over the cake {while still warm}. Allow syrup to soak in and continue until all syrup has been used.
Allow cake to cool completely in the pan {preferably overnight}.
Carefully remove cake from the bundt pan. This cake is rather delicate and you can not move it once it is on the cake plate.