Heat oven to 350°F. Oil a 2½- to 3-qt baking dish. In large bowl, whisk together eggs, milk, mustard, nutmeg and ¼ tsp each salt and pepper. Add bread and let sit, tossing occasionally, 5 minutes. Fold onion, apple, ham and Cheddar into bread mixture. Transfer mixture to prepared baking dish and bake until set and knife inserted in center comes out clean (cover bread pudding with foil if it browns too quickly), 45 to 50 minutes.