The benefits of -ate access, by type.
When talking about Special moves, I am mostly ignoring Uproar, instead focusing on Hyper Voice, Tri Attack, and Tri Attack-through-Nature Power. Uproar is just as strong as Hyper Voice of course, but does not represent a 'free' boost over using an equivalent-strength natural move, since it locks you in upon use. All other widely available special attacking Normal moves are 60 Power or less, and unlikely to represent an improvement unless the Pokemon is already sad and pathetic to the point of being near-unusable. (Such as Dedenne, which almost completely lacks Fairy moves)
It's worth noting that Arceus gets Hyper Voice. I presume it remains banned in this meta, but in any event Arceus should never run Judgment in this meta, unless you're concerned about Soundproof opponents. (You shouldn't be, really) After the -ate bonus, Hyper Voice is stronger, and it punches right through Substitutes.
All Eeveelutions get Hyper Voice, though that's not the same thing as saying all of them offer real value by doing so: Flareon has enough Special Attack to use this, Leafeon does not. Something to keep in mind though.
Anytime I name a specific Pokemon within a type, I have made sure to confirm that its primary type is the relevant one. By the same token, I have not listed examples like Meloetta learning Hyper Voice in Psychic, because I have already checked and determined that it's Normal/Psychic.
Normal: Normal types have somewhat less mediocre offense overall, but are handily outshined by other types regardless. In some cases it's even more dramatic than normal: Noivern pretty much invalidates Exploud as a Boomburst user, being faster, harder hitting, and with Flying type's only disadvantage compared to Normal being that Electric resists it. On the other hand, Mega Kangaskhan is even better than normal, since its primary offense actually is Return/Frustration.
Ghost: Physically oriented Ghosts get a huge boon: Shadow Claw, at a mere 70 Power, is their best single-turn attack. Return/Frustration is already nearly 50% stronger before the 30% -ate bonus, which places it just under twice as strong as Shadow Claw, the only loss being Shadow Claw's crit chance increase. So physical Ghosts have basically doubled their offensive prowess: Mega Banette in particular goes from having mediocre moves propped up by very good Attack to having a shocking amount of offense. Unfortunately many Ghosts are type/Ghost rather than Ghost/type, such as Golurk, holding them back in this particular case. Special Ghosts receive more limited benefit overall, having difficulty actually beating Shadow Ball with a Special Normal move: Gourgeist and Trevenant have access to Nature Power and therefore Tri Attack, and Mismaguius is notable for having access to Hyper Voice, allowing it to bypass Substitutes and inflict appalling damage generally, but no other Ghost is particularly improved in its special offense.
Dark: Physical Dark types don't see quite the same insane boost as Ghosts, since Crunch is widely available and 80 base Power, but there are a surprising number of Dark types whose best option is Night Slash, and for those Pokemon it's exactly as huge a boost as for Ghosts. Surprisingly, Dark special attackers are noticeably stronger overall, as a number of them have access to Hyper Voice, including the likes of Houndoom, Hydreigon, Zoroark (Combine with other Hyper Voice candidates to really mess with your opponent's head!), and if you're willing to use their bad Special Attack scores, Mightyena and Liepard.
Psychic: Psychic is in basically the same boat as Dark in terms of physical offense, and is actually even happier than Dark since Zen Headbutt has a miss chance where Crunch is completely reliable. Special-wise, Psychic can run Gardevoir is a somewhat ambiguous or even mildly dynamic threat: if it doesn't Mega Evolve it hits you with Psychic Hyper Voice, but if it does Mega Evolve it hits you with Fairy Hyper Voice. Additionally, Espeon and Mew get Hyper Voice. (And Chimecho, if you wanted to use it for some reason) Very few special Psychics really benefit from this meta, especially since a lot of them prefer Psyshock for special walls anyway.
Fighting: Unfortunately Fighting types more or less do not learn any Special Normal moves aside from some dubious moves like Round. On the other hand most Fighting types aren't Specially oriented anyway. Physically speaking, Fighting types get a nice boost to power without having to sacrifice accuracy. Overall most Fighting types would rather be type/Fighting than Fighting/type, as Fighting is one of the best offensive types in terms of movelist: Gallade, for instance, is exceedingly glad to be Psychic/Fighting instead of Fighting/Psychic, as swapping out Psycho Cut for -ated Return/Frustration nearly doubles its Psychic Offense where swapping out Close Combat for -ated Return/Frustration would be a less than 10% damage boost. (And also do away with the defense drops, admittedly)
Dragon: While not quite as dramatic as Ghost/Dark/Psychic, Dragon very much appreciates being able to replace Dragon Claw or Outrage for a move that hits harder than Outrage without its disadvantages. Special attacker wise, a whole bunch of legendaries can swap out Draco Meteor or Dragon Pulse for Hyper Voice, as can Altaria and Salamence. Other than those ones, Special Dragons don't really benefit.
Ground: Physically speaking Ground basically slaps a Life Orb over Earthquake, which is nothing to scoff at but is one of the weaker boosts. Earthsplosion on Landorus deserves special mention for being hilarious. Special Ground receives limited benefit: several Ground types have Nature Power, but none of them are Ground/type, and Dugtrio is the only Ground type to get Tri Attack, and has godawful Special Attack, and most Ground types oriented towards Special already have the solid Earth Power.
Rock: At last 100% accurate non-terrible-base-Power physical Rock moves! If only for escaping the miss chance, Rock benefits hugely. Special-wise, I think Diancie using Nature Power to get Tri Attack is the only case of a Rock Pokemon with non-terrible Special Attack particularly benefiting. If Snow Warning on Aurorus is ever released, it's something of an option, too.
Steel: Like Rock, the best physical Steel moves all have miss rates, so physical Steel benefits a lot for that reason. Special-wise Steel doesn't really benefit at all: the Steel types that can learn decent-strength Special Normal moves aren't Steel/type. Jirachi is the only one that so much as learns Uproar while being Steel/type!
Bug: Widespread access to stronger-than-Megahorn-without-missing? Yes please. Special attackers-wise, only a few Bugs can learn any of the three best moves, and even fewer learn Uproar! Genesect is one of the only ones to do so with worthwhile Special Attack, and is perfectly pleased to replace Bug Buzz with Bug Techno Blast, but otherwise special attacker Bugs are still not rising above Bug Buzz.
Grass: Physically speaking Grass is more held back by being a bad type than by a bad movepool, and also by the fact that most Grass types aren't physical attackers in the first place, but this could potentially salvage the likes of Cacturne. Power Whip users like Tangrowth are certainly glad to escape the miss chance, though. Special wise... well, Maractus gets Hyper Voice? And if you REALLY want to escape a dependency on Sun support AND wouldn't rather use Giga Drain, then there's no shortage of Grass types with Nature Power to acquire Tri Attack. But overall Grass gets comparatively little out of this meta.
Water: Since Waterfall (Or if you like missing, Aqua Tail) are the overall hardest hitting widespread physical Water moves, physically oriented Water Pokemon like Feraligatr definitely appreciate being able to nearly double their damage output on their STAB. Crabhammer users like Crawdaunt still experience noticeable increase in damage, and appreciate not risking misses. Special attacker-wise, Seismitoad, Vaporeon, Ludicolo, Politoed, Lapras and Wailord all have access to Hyper Voice and other than Wailord little reason to not use it, (Water Spout is still stronger at full or nearly full health, though I'd still consider Hyper Voice over it if I were using Wailord) but beyond that you have to fall back on Uproar if you want to bolster your Water types special attacking potential, which also only has a few potential users. Azumarill gets Hyper Voice if you want to run special Azumarill and don't mind that it doesn't get Moon Blast, I guess. It is worth noting that Manaphy is one of those Uproar users: it could potentially be worth running it in spite of being locked in by using it, at least as long as you don't see anything immune to Water.
Fire: Physical Fire types (Which is a lot of them) are often either basically doubling their damage output (If they would otherwise use Fire Fang or Fire Punch) or acquiring a smaller boost to damage but dropping recoil damage. (Flare Blitz) Sacred Fire users are the only Fire types that don't necessarily appreciate switching, and Entei gets Extreme Speed instead! Special attackers-wise, Pyroar and Flareon both get access to Hyper Voice, potentially redeeming Pyroar, while Typhlosion, Camerupt, and Heatran (And Magcargo, I guess) could conceivably replace their normal Fire STAB with Tri Attack-by-Nature Power. Victini is very glad to be Psychic/Fire, since it already has V-Create for ridiculous physical Fire offense, and Blue Flare for special Fire offense. (Its best Normal Special move is Uproar, though, so whatever)
Ice: Physical Ice of course appreciates replacing its various, mostly not-that-great physical moves with -ated Return/Frustration, or Facade, or whatever, some more than others. Special attacker wise, Glaceon and Jynx both bring Hyper Voice to the table, which is actually stronger than Blizzard with no miss chance (And no Freeze chance, admittedly) and throwing in bypassing Substitutes, and other than that... Special Ice mostly doesn't benefit, in short.
Electric: Physically speaking, there's not many Electric moves that don't suck, and the ones that don't suck are restricted in access (Fusion Bolt, for instance) and/or have substantial recoil. As such physical Electric Pokemon (Of which there are quite a few) experience a huge boost in their offensive utility. Special attacker wise, Magnezone gets Tri Attack and of course Jolteon gets Hyper Voice, and otherwise there's no real improvement. Potentially Rotom, Thundurus, Raichu, or Manectric could use Uproar, if you're willing to accept being locked in, but that's just asking for trouble.
Fairy: Physically speaking, Fairy in theory gets a solid boost and cheers at no longer missing on Play Roughs. In practice, Granbull and Clefable (Specifically, Clefable using Magic Guard and Double Edge, and maybe Belly Drum) are the only physical Fairies of any note that actually Pixilate Normal moves: Mega Mawile and Azumarill are both type/Fairy, not Fairy/type, and Slurpuff just switches from 'Play Rough with Return as "coverage"' to 'Return and hope to god nothing resists Fairy'. (Also, if you were to run physical Xerneas for some reason, it cheers at having, you know, any physical Fairy move at all) Special attacker wise, Hyper Voice is available to Togekiss and Clefable, in addition to the usual Sylveon and Mega Gardevoir usage, and is actually learned by a bunch of other Fairies with terrible Special Attack or that aren't Fairy/type (Not that this helps Fairy), and Florges (And Xerneas and, amusingly, Togekiss) can get Tri Attack through Nature Power. Not many special attackers benefit overall, but for instance Togekiss is faster, overall tougher, and slightly harder hitting than Sylveon.
Flying: Since only two Pokemon lines are pure or primary Flying type, this is largely irrelevant. Nonetheless, Noivern gets a fantastic Aerilated Boomburst, and physical Tornadus has a lot more potential to surprise people. You can give Tornadus Uproar if you want, but it, alas, acquires neither Hyper Voice nor Tri Attack/Nature Power, so its special offense is not meaningfully boosted.
Poison: Physically speaking Poison's straight damage is hugely improved, but it's debatable how much any given Poison type cares given their tendency to try to wear down the opposition. Special wise... Nidoqueen, Nidoking, and Crobat can all run Poisonous Uproar if you want? Poison is not particularly thrilled with this meta.