Take it From the Top
I think this question has some really specific nuances that can make it work. In order to arrive at the answer, I'm going to walk back from the question to arrive at a place where it becomes possible, and we will allow handwaving to make it plausible for your scenario. Since the question seems to focus on the Trihorners and Geckos (and since those species are the cool, new ones in this context) I'm going to focus on those, and fill in the gaps around them with the other pieces of the food-chain.
Strap in, this might take a while.
Large, Carnivorous, Active Reptiles
The two new species are large reptiles. As Franklin Pezzuti Dyer has already pointed out, the likely outcome of a nuclear war, nuclear winter, is very unfavorable to reptilian species due to the global temperature drop. Reptiles are ectothermic, so they require heat energy to be available outside of their bodies. However, there are two ways to deal with a lack of external heat: 1) Movement heat and 2) [Gigantothermy].
When a reptile moves it generates heat in its body due to friction and muscle activity. In a cold environment, physical activity can help offset a lack of heat. This, of course, comes at the cost of calories, so a more active reptile will need to consume more food. Reptiles do tend to be carnivorous or omnivorous in general, though, and if the insect population increases the reptile population can increase consumption in kind. Under this logic, a colder environment with enough available food will tend to produce replies that are more active and carnivorous.
A reptile's survival is still ultimately left to the whims of the climate, but that's where gigantothermy can help. You can describe an animal's size using both its volume and its surface area. As an animal's volume increases, its surface area increases proportionally less. This is known as the [square-cube law]. In practice, this means that larger animals have less surface area per volume than smaller ones, and thus are less affected by external temperature change. In mammals, this is not great, because we produce heat and need excess heat to be removed via our skin. Ectotherms, however, would find this very useful in colder climates, since larger versions would shed less heat than smaller ones over the same time frame, thus allowing them greater freedom in terms of the lowest temperature they could survive in. According to this principle, colder climates with enough food to go around might tend to produce larger reptiles.
Now we have explanations for the large, carnivorous, active reptiles, maybe we can assert that these reptiles come from specific current species. I first propose that the names be switched on the descriptions and that trihorners be the smaller version and geckos be the larger. The name trihorner evokes a sense of sharp, bony protrusions on the animal, and this seems to fit tortoises best as a precursor. Tortoises already grow quite large in general, and if they had to become more active and predatory to compete, the notion that they might adapt their bony shell into large protrusions is not too far-fetched. Add to that the fact that tortoises and turtles are generally omnivorous and I think you have a good match for this creature. Also, turtles are known to communicate with each other and learn better in groups, which supports the pack-hunting dynamic.
As for the gecko, which I propose to be the larger carnivorous animal, comodo dragons seem like the best precursor. They have claws, they are already quite large, and there's a bonus highly-deadly saliva weapon. They are a type of lizard that is very territorial and solitary and they dominate any ecosystem they live in. Also, comodo dragons are known to be able to produce offspring asexually from females, so if you were looking for a species that can go till the bitter end, look no further. As long as one female is alive, the species can continue. Imagine this creature, the size of a lion, crawling towards you at high speed because it needs more food.
Some Other Aspects Of Reptiles
Reptiles have an inefficient circulatory system as compared to most mammals, and when you couple that with the large body size what you get is a very short-ranged hunter. They won't be able to pursue their objective for long. This is fine for hunting animals that are probably dumb enough to get surprised easily or trapped, but humans would likely be able to outrun them easily if they could get any meaningful distance between them after the initial attack. Also, when energy is not needed, reptiles have a tendency to just sit and wait, using their excellent camouflage to hide from predators and prey alike while they conserve energy. They are also naturally danger-avoidant, so the smaller ones especially will try to run from a threat they are not ready for instead of attack unless running wouldn't work in the situation. That said, more aggressive reptiles wouldn't be hard to believe in this context.
Other Animals, what about them?
Literally all the animals are explainable due to humans keeping them while the rest of the world sorts itself out. Rats and bugs will hang out around humans and their settlements, functional or not, and the same is true with domesticated animals. The insects will likely breed out of control, as all of the answers point out, and this can be used as a food source for other populations. Also, it's worthwhile to note that radiation will, 99% of the time, just kill everything it touches. The idea that radiation will mutate things that survive is pretty ridiculous since those sorts of mutations will be cancerous and instead just kill everything. Have I mentioned that anything that gets heavily radiated will die? You can avoid this outcome by using fusion bombs instead of fission bombs, which produce way more blast power per level of radiation and should, therefore, leave behind way less overall radiation as a consequence. This also means a better short- and mid-term prognosis for this planet, as would not have to contend with the heavy fallout associated with fission bombs or "dirty" bombs.
Note: I know I need to add more links and information, but I'm going to save this now and come back to a little later. There will be further edits.