I'm picturing a world say a 100 years after a devastating nuclear war with a decades long nuclear winter where DNA damage due to war related radiation as well as subsequent ozone depletion have destabilized genetic lines. Some mutations have helped certain species. So not only I'm asking which species would fare better but also what would say a certain species may end up looking like. For example, we may see smaller cats with short limbs that can burrow deep underground, hence are half-blind with large mole-like claws, etc..
Contrary to believe, cocroaches would not survive a nuclear winter, but not because of the radiation, but because of the cold. A global nuclear winter would see the appearance of endothermic characteristics in animal that use to be exothermic. Sadly this would mean that lizards and insects would be much more voracious.
But the ability to lower the metabolism when there is little food or a creature finds itself in a warmer place could also be crucial. So many creatures could appear with the capacity to generate just enough internal heat to survive eficiently.
Well, for one thing, a decades-long nuclear winter is probably not a real thing (at least not with any quantity and size of nuclear weapons humans have made so far). The idea of apocalyptic nuclear winter in the first place is quite questionable.
However, assuming it was, we're basically talking about something like the end-Cretaceous extinction event, but maybe worse, and with radiation.
So good adaptations:
cryptobiosis: survive through the nuclear winter in an inert state. Tardigrades are famous for this, but other small invertebrates can do it too - some rotifers, some nematodes, at least one insect. https://en.wikipedia.org/wiki/Polypedilum_vanderplanki - "In the dehydrated state the larvae become impervious to many extreme environmental conditions, and can survive temperatures from 3 K to up to 375 K, very high (7000 gray) levels of gamma-rays, and exposure to vacuum."
This is pretty much the perfect adaptation for surviving something like this, but it doesn't seem to be available to vertebrates. (Well, the wood frog can freeze solid and recover, but it's only known to do this over winter IIRC, not for multiple years.)
feed on something not dependent on sunlight: deep-sea tube worms etc. whose food chain is based on chemosynthesis rather than photosynthesis. Again, this is only available to a very limited set of animals.
eat pretty much anything, including dead stuff: small omnivores that can scavenge probably have the best chance among vertebrates of getting through a mass extinction. While I can't remember where I read this, IIRC freshwater streams were less badly hit by the end-Cretaceous extinction since the ecosystems are largely detritus based.
Radiation probably won't be terribly determinative of what survives and what goes extinct, since most of the most vulnerable things will probably go extinct anyway from the food crunch. Many invertebrates are really resistant, and even among the more vulnerable mammals etc., there will probably be pockets where the radiation levels are lower.
If radiation is an issue, I believe cockroaches are more resistant to it because their cells divide less frequently than other organisms, so a lower cell division rate might help.