I love questions like this! Here's some ideas I came up with. (And this video is also an excellent watch to learn what to avoid in this genre, so I recommend that too.)
My advice, in brief, is this:
Either avoid the cliches of the nuclear apocalypse genre, or twist them in a creative way.
You've seen Mad Max, Fallout, and similar media, and a lot of that media tends to share certain common ideas about a nuclear apocalypse setting, to the point where many of them do it almost exactly the same way. This isn't necessarily bad - after all, tropes are tools - but if you're trying to differentiate your setting from those settings, these common traits might be some things to avoid. You're trying to make your story unique, to put your own spin on things, and that means taking the ideas from those who came before you and twisting them, or telling them from a different perspective.
Here are some typical nuclear apocalypse genre cliches, and how you can either avoid them or put your own spin on them.
A tough-as-nails, gruff protagonist with no clear motivations.
The "Sole Survivor," the "Wanderer," the "Mariner" - all of these are just names for the protagonist of a nuclear apocalypse story, and that protagonist tends to have very specific genre cliches. They will be gruff, hard, super tough, and they might drive a cool vehicle or have lots of weapons to defend themselves. They wander the wasteland in search of... something vague. Revenge? Greed? Sometimes that's not even clear. And they're tough. That's the most important thing. They're also typically a grizzled, unshaven white male in their mid-thirties or forties, maybe with a dead wife or a missing or dead child - what Yahtzee of Zero Punctuation succinctly calls a "hairy dad."
It's a little boring, isn't it? That's because it's been absolutely done to death.
In order to put a spin on the usual genre cliches, you could have your protagonist deliberately spurn some of these traits. Instead of being a gruff, no-nonsense man of few words, what if your protagonist was a scientist who survived the apocalypse through technological prowess and using their knowledge and book smarts to purify water and find food, a la The Martian, rather than just generically "being tough"? What if instead of being super paranoid and liking to work alone, they were soft and friendly and helpful, trusting others and being generally warm and likable, like the Mandalorian? There are so many ways to be creative with an apocalyptic protagonist beyond being the generic Sole Survivor type, so embrace it!
Barren, desert wasteland with ruined cities, desolate hills... and nothing else.
The vast majority of nuclear apocalypse settings in media have the world be a hot, scorching barren desert wasteland dotted with the ruins of cities. It sounds interesting in theory, but it often just serves to have characters walking across vast boring stretches of empty land, and scavenging around for resources - which is hard to write in an engaging way, because it's just not very interesting. How often in Fallout do you just walk around giant areas with nothing to do or see? Why not mix up your wasteland and make it more than just a generic desert?
Some of the most engaging missions in Fallout games don't take place in the wasteland - rather, they take place inside abandoned robot factories where all the robots have gone bonkers and try to kill their guests, or inside a diner run by a weirdo druggie lady whose son is hooked on chems. The best stories in your apocalypse setting will come from the interesting places and landmarks that populate your wasteland, not the wasteland itself. Focus on that.
Alternatively, why be limited to just a barren desert wasteland at all? Waterworld did a fantastic play on a post-apocalypse setting by having the world be flooded by the melting of the ice caps, making for a crazy, weird and cult-classic high-seas adventure that is still to this day one of the most unique apocalypse settings ever made.
Mutants and radiation powers.
You know what I'm talking about - the thing in basically every nuclear apocalypse setting where the vague force of "radiation" and "radioactivity" has warped animals and people into crazy mutants. The cockroaches are giant now and they glow! People have turned into skinless ghouls or three-eyed hunchbacks! Wild stuff! Except that the concept has been sort of done to death, and it's not at all how radiation works in the real world, so in order to make this interesting you'd have to really shake things up.
Instead of going in the typical, "crazy science" direction, why not go in a realistic direction and portray what actually happens to people who are exposed to radiation in the real world - that is, radiation poisoning, leukemia and cancers? That could make for strong human narratives. Maybe you could have a side plot where one of the characters gets genuinely ill from radiation sickness when they first go out into the wasteland and are exposed to radioactive debris. That would be far more engaging and interesting than just "wow, Timmy grew three eyes!"
Heavy metal band survivors, outlaws, anarchists, and bandit gangs.
These survivors are tough and nasty, man! They wear crazy biker clothes, have mohawks, and wear tons of neon pink for some reason, and they all ride motorcycles and drive around the wasteland in search of gasoline and people to rob, or something. They don't follow the rules!
...Except that literally nobody in a real apocalypse would ever act this way if they were smart. Robbing people and trashing places is not a good way to survive, and "anarchy" and "doing whatever you want" isn't conducive to being in an organized crime syndicate with strong goals and hierarchies. Your bandits wouldn't be able to predate enough or cooperate enough to survive, and your "tough as nails" outlaw renegade gang would quickly die out. Think about gangs in the real world. Are they "wild" and "lawless", or do they have hierarchies and rules just like any other society? As human history has shown time and time again, cooperation and strong leadership always triumphs over anarchy and banditry. If you're going to have bandit gangs and outlaws, don't make them wild psychopaths with no morals or goals - make them dangerous and organized. That makes them both more interesting and more realistic, and turns them into a real, genuine threat to your protagonists.
Ask yourself these questions while writing bandits and outlaws into your apocalypse narrative:
Why would somebody want to join this gang? What does it offer? The answer should be more complicated than just "freedom" or "power" or "stealing stuff."
What is the hierarchy and social structure in this gang? Is there a "boss," or are there multiple commanders who keep the peace and curtail all these unruly gangsters into a coordinated force?
What are their unique traits that distinguish them from other groups in this setting? This should hopefully be more complex than just a cool outfit or a specific catchprase or haircut.
Zombies, undead, infected, walkers, Zeds...
Whatever you call them, so many apocalypse novels and movies have done zombies to death that you'd really have to put a new spin on them to stand out. How will your zombies be different from the generic zombies we see all the time in pop culture? What's cool or especially scary about them?
The Last Of Us did something new with zombies by having the zombies be fungal, and basing them on the real-world Cordyceps fungus, giving them an interesting basis in science and a unique visual design. Instead of just being generic dead people, the zombies had glowing mushrooms and fungal plates sprouting eerily out of their faces and bodies, and each zombie type made biological sense - "clickers" were zombies whose fungal growths covered their eyes, so they adapted to hunt by sound. The game also explored the human aspect of zombies deeply, and had characters discuss them philosophically, asking whether there are humans left inside of them and whether it's ethical to kill them.