Mix peaches, flour, 1/4 cup white sugar, rum, 1 tablespoon cinnamon, and 1 pinch nutmeg together in a baking dish.
Bake in the preheated oven until peaches are tender and bubbling, about 30 minutes. Remove and set peaches aside.
Reduce oven temperature to 350 degrees F (175 degrees C). Grease a 1 1/2-quart casserole dish.
Stir milk and heavy cream together in a saucepan over medium heat; heat until just warmed and a film forms over top of mixture, about 5 minutes. Remove from heat; stir in butter until melted. Cool mixture to lukewarm.
Combine 1 cup white sugar, brown sugar, eggs, 2 teaspoons cinnamon, vanilla extract, salt, and 1/4 teaspoon nutmeg in a bowl; beat with an electric mixer at medium speed until combined, 1 minute. Slowly pour milk mixture into sugar mixture; stir to combine. Add bread cubes; stir gently. Fold in cooked peaches.
Pour batter into prepared casserole dish.
Bake in the preheated oven until pudding is bubbling and center is set, 45 to 50 minutes. Serve warm.
