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.