I don't think that it would transfer. Most diseases are only spread through one species, and things like avarian flu or swin flu are mutations. Most predators would not eat a zombie because they do not like rotting flesh. The only animals that would dine on the dead are scavengers, and they unique digestive systems that can digest rotting foods, so I do not think we would see zombie animals. Just an increase in scavengers (cyotes, crows, vultures)
However since you are wanting to fear animals as well, there would be escaped animals from zoos, packs of wild dogs, etc.