I'll step through each of the classes individually, but first the broad strokes:
[Most] Spellcasters will fare much better than everyone else
The main check on the power of a Spellcaster is their limited resources. If a Level 9 character uses a 5th Level Spell Slot, that's it: that's the only fifth level spell they'll get for the whole day. Wizards and Land Circle Druids can recover some of their spell slots, once per day, on a Short Rest, but in general, once spell slots are used, they're gone for the day.
This means that if a spellcaster enters combat for the day and does not expect any other combat for the rest of the day, they can blow their most powerful spells with impunity, and not really think about conserving them just for the possible scenario that they might need a really powerful spell later.
Conversely, classes that don't have resources, or have resources that recharge on a Short Rest (most Martial Classes; also Warlocks) will not gain this benefit. Their resources are designed to be expendable in this manner in normal circumstances, so being able to do so in a "Five Minute Adventuring Day" does not benefit them—and from their perspective, the change will feel like a Nerf, since they'll probably go from using their features 4-12 times a day to 1-3 times a day, depending on the feature. Concentrated into a single fight, that may not be such a big deal, but it can make those features more precious and more valuable, making them harder to justify using.
Barbarians nominally benefit from this playstyle. Barbarians get a limited number of Rages per day (before level 20) and each Rage lasts a minute (longer than the typical duration of a non-boss fight).
Because a regular Adventuring day as defined in the Dungeon Master's Guide will last about 6-8 Combat Encounters, a Barbarian would only really be able to Rage for 20%-60% of those encounters, depending on their level (lower for lower level characters, higher for higher level characters). And if they drop out of Rage due to missing damage or for some other reason, it's potentially discouraging to try to refresh the Rage.
In the Five Minute Adventuring Day, they can stay enraged 100% of all time they spend in combat, and feel a lot better about Refreshing Rage in the situation where their Rage drops on accident. That's pretty good, offering a substantial reduction in overall damage taken, but only a marginal improvement to damage output.
Still, of the Martial classes, they probably benefit the most substantially.
There's not a lot to say here: Bards are full spellcasters, so they gain a tremendous benefit from this playstyle. There's not much benefit to saving the Dominate Person 5th level spell slot if you know that there's not much else in the day, and dominating a critical enemy creature might drastically shift the balance of the fight.
One thing worth pointing out though is that Bards do have quite a few features that recharge on Short Rests, like Bardic Inspiration (after level 5) and Song of Rest, the latter becoming nearly completely useless in this kind of playstyle. So a Bard might feel nerfed compared to other spellcasters because of this. But overall, they're still generally benefitting.
Same deal as a Bard, except swap Bardic Inspiration/Song of Rest with Channel Divinity.
For a Cleric, the Channel Divinity nerf might feel harsher, but it's still the same overall picture, with their Spellcasting features standing out more prominently.
Same as Cleric, swap Channel Divinity with Wild Shape.
Moon Druids are going to feel less inconvenienced than other Druids, just because their one Short-Rest feature, Wild Shape, is going to feel more consequential than it would for other Druids. But otherwise, they're mostly going to enjoy the ability to spam high-level Moonbeams or Healing Spirits.
This is going to depend a little more on the Archetype, but in general, Fighters are the first class where this playstyle hurts more than it helps.
Action Surge and Second Wind, both features that recharge on Short Rests, will now only be available once (or for Action Surge at later levels, twice) per day. For most Fighter Archetypes that have expendable resources, like Battlemaster and Arcane Archer (and on technicality, the Banneret archetype), their archetype features also recharge on a Short Rest, meaning they'll only get those features a minimal amount of times in a day. That will feel like a substantial nerf.
Conversely, the Cavalier and Eldritch Knight archetypes will feel a little better, with their Long Rest-based resources being more freely usable, but the Cavalier tends to see only niche play, and the Eldritch Knight's spellcasting is only 1/3 that of a normal spellcaster, so it'll be much less substantial than a regular spellcaster.
As Monks' only major resource is their Ki points, which recharge on a Short Rest, Monks will be hit the worst by this playstyle. Monks have no features that recharge on a Long Rest, and even of the archetypes, the only Long-Rest based feature available is the Open Hand Monk's Sanctuary effect, whose effect in combat is minimal.
So whatever Ki points a Monk gets at their level, is all they get for the whole day. They can spend them very quickly with judicious use of their features that consume Ki points, but none of those features are as game-changing as the spellcasting that other classes get, at least before the highest tier of play.
Paladins probably benefit the most from this playstyle, excepting maybe Sorcerers. They're only half-spellcasters, so they can't spam out max-damage spells with the same deadliness as other spellcasters, but they do have Divine Smites they can more freely deal out as they fish for Critical hits, and a Paladin who times their smites well can brutalize many encounters. And of the spells they do get, Paladin spells often hit way above their paygrade for the level of slot they consume.
Like Clerics, they're going to suffer for not being able to use their Channel Divinity more than once a day, but also like Clerics, they'll appreciate being able to spam out their most powerful features otherwise.
Rangers are ostensibly in the same boat as Paladins, but they lack the especially powerful smiting abilities that Paladins have, and some of the most powerful Ranger features don't have resource costs attached, so their advantage is less pronounced. They will enjoy getting to use their most powerful Ranger spells with wild abandon, though.
Strangely enough, Rogues generally don't have resource costs associated with their features. Assassins have a de facto limit on their features because they generally only activate on Surprised creatures (which can only occur once per combat), so they will feel a nominal nerf to their abilities, but other than the Arcane Trickster (which can be lumped in with the Eldritch Knight as receiving a nominal buff), Rogues won't feel many changes from this playstyle.
Sorcerers rival Paladins in terms of power gained from this playstyle. Not only do they have the same benefits as other Spellcasters in gaining the ability to spam out their most powerful features with wild abandon, Sorcerers also have features like Twinned Spell and Quickened Spell (or Font of Magic) that will let them use their Action Economy more efficiently, pumping out their deadliest spells faster and more dangerously than other spellcasters can. I think it's fair to argue that Sorcerers benefit the absolute most from this playstyle.
And unlike other spellcasters, their only major Short Rest based feature is gained as their Level 20 Capstone ability and In mY OpInIoN, it kind of sucks anyways, and is easily one of the worst level 20 capstone features, so they're not even going to feel any nerfs from this playstyle. For Sorcerers, this playstyle is pretty strictly all upsides.
Wizards have the Arcane Recovery feature, so they'll experience a small nerf compared to other spellcasters, but otherwise this playstyle continues to be mostly upsides, with the gained ability to spam out powerful spells with impunity, attached to a very large Spell List that gives them a tremendous amount of flexibility.
Warlocks will feel a heavy nerf in this playstyle. It won't be as bad as it is for Monks, because Warlocks do gain quite a few Long Rest features, whether from Invocations, Archetype features, or Mystic Arcanums, but their primary spellcasting will only permit a few spells a day, only up to four even at level 17. That simply will not favorably compare to a Wizard or Sorcerer spamming out their highest spells, especially considering the very limited list a Warlock has to choose their Mystic Arcanums from—and it should not be left unstated that the Mystic Arcanums only start coming online after level 11, meaning they won't benefit players in the level range that spans the majority of campaigns.
||Effect of the Five Minute Adventuring Day
||No Impact, maybe Minor Nerf