Bread doesn't cause an allergic reaction, it has a knock-on effect. It is low in nutrition and high in bulk, swelling inside the hog’s tummy making him feel full, to the detriment of further foraging to fill up on the right food stuffs.
The old myth of feeding milk and bread to hedgehogs is potentially very harmful indeed. Hedgehogs are lactose intolerant, and drinking milk can lead to diarrhoea, dehydration and death.
For suggestions of nutritional supplementary foods to feed hedgehogs, see
here.