@LukeAlmighty @Terry
I think you're taking the worst case possible, and assuming hydrogen bombs will literally destroy everything. In reality you'll only have bombs drop on major cities, with smaller ones and rural areas remaining unscathed. You'll still have farms capable of producing food, you'll still have plenty of domesticated animals.
The problem arises if you had fissile nukes drop instead, cause the radiation would spread even over the areas not attacked directly, and this would be the case where even farms would be compromised and you no longer have a good food source.