Bread pudding doesn't have to contain yeast breads. A recent Chowhound discussion focused on recipes built around quick breads and cake.
Day-old quick breads (banana and pumpkin, say) are popular. By leaving the oil or butter out of a quick bread, it'll better absorb the custard in a pudding recipe, chowser says.
Stale pound cake works well in bread puddings with berries, team_cake says. Emme likes bread pudding that combines carrot cake with a buttermilk custard. Leftover scones, muffins, and even pancakes can all work in bread pudding. And day-old cornbread and buttermilk biscuits work well in both sweet and savory puddings.
Discuss: Alternatives to yeast bread in bread pudding
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