How I'd do it. Long post incoming...
Races - reskin half-orcs as muls, except replace Savage Attacker with something more defensive/endurance based. Half-giants need a bit of a boost over the rather underwhelming goliath, but size L is just too difficult to balance with other PC races. Kreen would be required, of course. Elves lose racial weapon proficiencies and cantrips, gain an increased movement speed. Dray are breath-weapon-less dragonborn with darkvision.
Character creation - I'd allow all PCs to take a feat at first level, as a sop the the 2e days when Dark Sun PCs were Just Better. Wild Talent is a feat, so if you want to be one, this is how.
Classes: no spellcasting bards. Remove the class entirely or else provide an alternate that replaces all spellcasting abilities with leadership/negotiation based abilities for military leaders or dune traders (and alter proficiencies accordingly). Four new cleric domains, one for each element (you could get away with using Light as Fire if you have to), templar clerics can use existing domains depending on their sorceror-king. Paladins can stay but may have the tenets of their oaths rewritten to better fit the setting. No arcane tricksters or spellblades. Perhaps a new rogue subclass specialising entirely in poison? Only warlock patrons available are the sorceror-kings. No sorcerors, at all, unless this is how you decide to do psionics.
Templars: drawing heavily from Lynn Abbey's take. Sorceror-kings can hand out templar medallions to anyone regardless of class, these are minor magic items that allow a minimal level of communication between wearer and sorceror-king (and one-way monitoring), not all templars are actually capital-T templars mechanically (this is a part of the new 'Templarate Member' background but also can be gained in play if a PC wants to enter the service of a sorceror-king). Sorceror-kings can empower both cleric and warlock templars, and these two classes get a medallion as their holy symbol/arcane focus.
Psionics: I have no preferred implementation at this stage, but for me, psionics should 1) require no material or verbal components, 2) require no preparation, 3) use a unique and traditionally 'psionic' spell/effect list, not just blindly use everything off the sorceror spell list, 4) be able to choose any mental ability score for their casting, not be locked into a single one. Wild Talents should have an actual psionic effect that they can use, not just a psionic dice like the UA feat chain gave them.
Wizards: defiling should be a temptation for all wizards, it shouldn't be defilers/preservers as subclasses. I'd make 'Defile' a bonus cantrip that every wizard gets at first level, is usable as a bonus action, and boost the next spell cast (or remove a setting-wise penalty to wizard spellcasting, alternatively). Not sure exactly what benefits it gives, but they'd have to be noticeable. Or perhaps defiling could be the default spellcasting method, and 'Preserve' could be a bonus-action cantrip that just cancelled out the destruction that would otherwise have occurred. Problematic with reaction/bonus action spells though? Hmm. would require playtesting. Wizards should have access to Deception and Sleight of Hand as proficiencies, to help conceal their spellcasting.
Equipment: Non-metal equipment is default and uses standard stats. Metal armour can just use the adamantine rules, Metal weapons need a similar simple universally applicable bonus, not sure exactly what, perhaps just a non-magical +1 to hit or damage, along with advantage on weapon breaking checks? While we're on the subject, metalsmithing tools should be a separate proficiency to non-metal weapon/armoursmithing tools. It's a small change, but in Athas metalsmithing skills should be as hard to come by as metal itself. Most of the funky Athasian weapons can just be implemented as reskins of existing weapons, perhaps with a damage type change - i don't think we need special rules or unique profiles for EVERYTHING.
Spells: any food/water creation spells need to be nerfed into the ground. Same with planar travel spells.
Setting: I'd put it in the same timeframe as 4e Dark Sun, after the death of Kalak. Tyr is now analagous to the reconstruction-era USA South, with the slaves nominally free under a theoretically democratic government, but largely still in thrall economically to the property-holding elite who use dominance in politics and night-time terror tactics to preserve most of the benefits they had under the old system. Some of Kalaks's templars still get spells (actually empowered by Sasha and Wyan) but keep it very secret.
The theme of the setting is survival and doing good in a bad place. The Sorceror-kings are evil, but also human. They're old, jaded, capricious, bored, cruel, arrogant, steeped in ancient rivalries and resentments, and hold life very cheaply indeed - they're not, as a rule, cackling psychopaths any more than they're idiots. Also, they DO provide security, food, water etc for the inhabitants of their cities, the old security vs freedom dilemma. It's not necessarily the only possible system, but it's the one that works now, and lots of well-meaning people might defend it out of fear of the alternative. PCs may choose to confront or try to overthrow the sorceror-kings as a long-term campaign goal, but that's for the level 20 range, and that's not the be-all and end-all of an Athas campaign. Founding and defending a free village or tribe, or a trading concern, opposing a particular noble house or slaving ring or powerful evil templar/defiler/psionicist or undead creature are other possible campaign goals. Small goods are worth doing. Doing all this while remaining good in a world where tyranny and slavery is the default mode of existence is the challenge. How much do you tolerate among your allies and those you make common cause with? How much will you compromise? What will you do for survival? Athas kills its inhabitants all the time and you can't help them all - who will you and won't you help, and how do you decide?
Thematically, Athas is THE 'social justice' D&D world. Slavery is front and centre and an evil to be fought. Environmental devastation is rampant and unambiguously the result of greed for power. There's a history (a long time back, but it's there, and there's still the odd creature around who remembers it) of humans trying to exterminate every other race, often successfully, to create a 'cleansed' world. Law enforcement is profoundly brutal and serves at the whim of untouchable tyrants (even if you joined the Urikite templarate with high principles, intending to patriotically defend your city against crime, and water theft, and defilers messing up the city's grain fields, and psurlon cults, you still serve a monstrous ancient inhuman control freak with a history of successful genocide who demands absolute obedience to every command, and some of your fellow templars are the worst people imaginable but your king doesn't care at best and will take their side at worst). 5e DS should emphasise that opposing any of these (let alone all at once) is HARD.
Geography-wise, I'd try to expand the world a little beyond the Tyr Valley region. Rewrite the kreen grasslands to be more ecologically hostile, to explain why the sorceror-kings haven't moved there en masse long ago. Detail the Dead Lands to the south a bit more. Maybe make it a bit easier to get to Ur Draxa (even though it's very, very hard to stay free or even alive once you're there, much less leave the place!) Talk a bit more about Daskinor, and the ruins of Kalidnay, and allude to myths and rumours of other sorceror-kings in other continents and parts of the world (I once messed around with converting the PF Rise of the Runelords AP to Athas, making the titular runelord instead a forgotten sorceror-king called Karzoug Gnoll-Eater...). Take the goofiness out of the Last Sea region, the Orwellian happy-tyranny under insane psionic echoes playing out a millennia-old love triangle is fine, just make the water too salty to drink and get rid of the surfers and pacifist lizardmen. Oronis and Kurn is ok, but needs to be a cautionary tale. Oronis renounced defiling and rebuilt himself and his city, but the transition was long and painful and involved great suffering as Kurn was ruthlessly exploited by outside forces while its king was indisposed. Even now, there's unhappiness and people who resent Oronis for abandoning them for so long while he chased some self-indulgent dream, and Oronis himself is continually torn between allowing his people freedom and democracy vs the urge to intervene when he sees them making self-interested or ignorant decisions. And what does Oronis do to comply with Borys's tithe, anyway?
I think we're vanishingly unlikely to see any sort of campaign or adventure book for Dark Sun even if the setting gets updated, but in the case we do, then really the archetypical Dark Sun campaign would be 'the Prism Pentad except with the PCs as heroes this time around'. Complete sandbox. Start at level 5 or something, in Tyr or one of the villages surrounding it. The triggering event is Borys's demand for a tithe of slaves. The PCs become aware of this either through being connected to the new Tyr government when they receive Borys's demand, or else being the victims of one of Tithian's clandestine slave raids as he tries to fulfil the tithe without the rest of the Tyr council knowing, or even being emissaries of another city-state who are in Tyr and who have been instructed to ensure Tyr complies (cos even the other sorceror-kings are afraid of what Borys - or even Rajaat - might do if they don't). After this - sandbox based on player choice. Detail where the PCs might find out scraps of the lore of the true history of Athas - Kemalok, the Pristine Tower, Sasha and Wyan, some ancient undead creatures, even the living Sorceror-kings or the ruined fortresses of the dead ones. Tithian as interim villain (if he kills off Sadira, Agis, Rikus and the rest then that leaves the floor open for the PCs). Detail finding the Dark Lens and what it can do, or the Scourge. Detail what impacts the PCs choices might have - if they refuse the tithe, then maybe one of the other sorceror-kings will do it for them (Albeorn did it in the PP i believe) but they'll be out for restitution (and blood) after, so war is likely. There should be temptations for the PCs to become Rajaat's witting/unwitting instruments, either via transformation in the Pristine Tower, or by becoming a proxy sorceror-king via Sasha and Wyan. And at the end of the day the devil's choice between Rajaat and genocide, or Borys and tyranny, unless the DM wants to be a big softy and allow them to find a third way out. Bonus points for getting PCs to betray each other or take different sides in the final confrontation...
