Combine the carrots, apples, sultanas and breadcrumbs in a bowl. Sift together the flour, baking powder, cinnamon, bicarbonate of soda, salt, nutmeg and allspice in a separate small bowl. Cream together the butter, sugar and eggs in a separate large bowl; stir in the flour mixture. Add the carrot mixture and mix well. Transfer mixture to a lightly-greased, 2-litre mould.
Place a steamer rack in the bottom of a large, deep pot. Set the filled mould on top of the steamer rack. Pour enough water into the bottom of the pot to cover the bottom 2/3 of the mould. Bring to the boil over medium heat. Reduce heat to low; cover. Simmer for 2 hours 45 minutes, adding water occasionally to maintain the water level. Remove from water and allow to cool 10 minutes before turning out of mould onto a plate.