Chickens, tilapia, channel catfish, goats, llamas, and maybe turkeys
If you survive the nuclear apocalypse, the first thing you're going to worry about is food. You need a food source that's fast-reproducing, easy to breed, can feed on what little you can find in a post-apocalyptic environment, and can be raised easily. Protein is going to be a big issue, since you can easily store plant seeds. Protein rich plants like soybean and quinoa, or other options like spirula might be hard to obtain in such an environment. You need something that's low maintainance and can be taken just about anywhere.
Answer: I present to you the humble chicken. Chickens...
- Breed fast, they reach maturity at about 16-24 weeks of age. Broiler chickens can be slaughtered for food at 7-9 weeks
- Eat just about anything, chickens are omnivores and will eat insects, nuts, seeds, berries, centipedes, etc. If insects survive to any noticeable degree you can find enough food to feed chickens. Chickens can survive pretty much anywhere outside of the polar circles.
- Can be easily handled. This is a big deal. Horses, cattle, and a lot of large mammal require a lot of husbandry and can be difficult to manage. Think trying to range cattle without having access to horses to herd them or the tools necessary to slaughter and butcher them efficiently. Chickens can be easily herded and killed if necessary. You can kill them by hand, or with a shovel or broomstick if you get a big, aggressive rooster.
- Can produce food without dying (eggs). This is also very important, as it means you can produce food for people without reducing the very limited head of livestock you have at any given time. Even insects aside from honeybees don't do this on a scale efficiently enough for human consumption.
- Are easy to raise. Human farms since time immemorial have raised free range to semi-free range chickens. In a post-apocalyptic scenario you might even have an advantage because there are no foxes or coyotes to eat your chicken.
- Don't require a lot of space. You could technically raise chickens within the confines of a nuclear vault, provided you had sufficient grain. You couldn't do that with sheep, horses, or cattle.
Turkeys are kind of similar, being big chickens if you oversimplify things. However, they aren't as space efficient and can potentially be a little more dangerous. But they have a lot of the same benefits (omnivorous, breed easily, etc.)
Other animals that fit this criteria include tilapia (you can literally raise them in sewage tanks, they're easy to breed, and good to eat) and channel catfish (also easy to raise), as well as goats (a bit hardier than the other major hoofstock). Goats can also produce food without having to kill the animal, and you can also get leather out of them.
Sheep are a bit of a trap, they're originally a mountain species that will survive in the harsh conditions of a wasteland and they produce a very useful substance in the form of wool, but those same adaptations that let them survive in harsh environments will lead them to strip the surface cover from your meager post-apocalyptic soils and send you into a dust bowl, something you probably can't survive on top of a nuclear apocalypse. If you absolutely need wool get a llama, they aren't as destructive and you can use them as a pack animal on top of that.
Horses would be very useful, but are a huge amount of work to feed and maintain. Cattle just don't even bother, the downsides outweigh the benefits.