Preheat the oven to 350 degrees F (175 degrees C). Butter a 9x5-inch loaf pan.
Mix bread and raisins together in a bowl; transfer mixture to the loaf pan.
Beat eggnog, eggs, white sugar, and vanilla extract together in a separate bowl. Pour over the bread mixture.
Bake in the preheated oven until a knife inserted into the center comes out clean, 40 to 45 minutes.
Melt brown sugar and butter in a saucepan over medium heat. Add cream and rum; simmer until sauce thickens and is reduced to 1 1/2 cups, about 5 minutes. Serve warm over the bread pudding.