We won:
Aerodactyl, Granbull, Torkoal
We lost:
Arcanine, Froslass, Kingdra, Venomoth, Zapdos, Drought
Here's a few predictions for how shifts may affect the tier:
Winners

The loss of Drought, Zapdos and Arcanine are all huge for
Sharpedo, as RU lost a weather weakening its Water STAB, the sole bulky Electric and Shark's most reliable revenge killer. I was on the fence about Sharpedo before, but these losses make it very scary to face unless you have a) bulky water/water immunity or b) priority user, though most cannot freely switch in. But not just any mon will do: for the former, you need to have reliable recovery or resist Dark STAB, and if you go the latter- oh yeah, Shark resists most common priority and blanks Fake Out with Protect. Before factoring in the gimmicks, your options with those criteria are:
A): PD Mantine, PD Vaporeon, PD Gastrodon, Poliwrath, Toxicroak
B): Hitmontop, Hitmonlee, QA Swellow, QA Zangoose, Linoone, Toxicroak
Not the worst list of checks, but bar Croak and some Top builds none of the priority users really take hits well in the event Shark predicts and hits them on the switch. Croak resists Dark and blanks Water and can then revenge with Vacuum Wave, but it drops to potential EQ. That said, Intimidate Granbull can effectively pressure physical Shark, so we may see Shark use special sets more in response, which means more Ice Beam and is why I don't count Grass types as good Shark answers. The recent departures make Shark more oppressive imo, making it more likely we might see a suspect.

With Zapdos gone,
Rotom and
Raichu are now the only truly good Electric types (sorry, I'm not using something that flops against Ground types even worse than Zapdos did). Rotom still packs a really good STAB combo and three immunities, making it a very good Choice user; Raichu sets itself apart by directly threatening Ground types bar Torterra with coverage and using its terrific Speed tier to get off plot.

The loss of Fross means there is a new opening for a dominant Spikes Setter, and of the candidates I feel
Omastar and
Cacturne are in the best position to take that title. Omastar is the new best setter for HO thanks to Weak Armor, allowing it to get multiple layers, and it also has access to Rocks, Toxic Spikes, Scald and the still rare Knock Off, which allows Omastar to get a lot of early game utility as a suicide lead; it can even go more offensive with Ice Beam. Cacturne isn't as good, but it packs several advantages as a spiker, namely STAB priority and two immunities, the latter of which actually gives it some use outside of the lead slot.

Hazard removal was not too good in the Alpha days, both because of the HO nature of early metas and the presence of one particular mon: Sableye. Sableye can shut down any remover with its Prankster antics and Ghost typing besides the niche Skuntank, now niche Shiftry and the Knock-weak Xatu, and you'd think it'd be better since the strongest Defogger just left. Enter Granbull; yes, Granbull doesn't appreciate burn + Taunt, but it can potentially overcome these with Heal Bell or RestTalk, with Sable struggling to make much progress against RestTalk without risking a healthy Play Rough as it Taunts; wake up turn is especially dangerous for Sable.
Xatu's Magic Bounce can further alleviate any Sable issue that Granbull may face, its Psychic STAB also complemented Granbull's offenses. Granbull will likely be at least decently huge, and that dip for Sableye along with the meta calming down leads me to believe that many removers will actually have an easier time, especially since several spinners also complement Granbull offensively;
Claydol gets around physical walls and Poisons with its Plot set, while
Top or Lee can pick off frailer offensive threats that can overwhelm Bull due to its low Speed. May not be meta defining, but I do believe these two factors will give at least a slight indirect buff.

Despite an EQ weakness,
Toxicroak is still probably one of the best checks to titan Sharpedo between Dark resist and Dry Skin; Croak is honestly a good mon in its own right, the only Fighting type that can viably run a specially offensive set, which along with Poison STAB allow it a favorable matchup vs Granbull. With the meta calmed down, I think more players will start to catch onto its advantages.
Losers

Lumping these two together as
Lapras and
Bastiodon both face a similar issue; they were mainly used because they were really good at countering one specific mon each, and now those mons are gone. Lapras' lack of good recovery and Bastio not providing much anymore besides a Boomburst sponge (which Mime does better since isn't weak to always present Focus Blast and can directly threaten Exploud with its own Blast) mean both are mostly outclassed by other options.

While Granbull's introduction will probably cause issues for Sableye (as explained above), the gremlin does have the tools it needs to persist;
Spiritomb has it a bit harder. RestTalk Granbull is extremely threatening to the current Tomb set, as it needs quite a few boosts before Dark Pulse can really break Bull while Bull naturally speed creeps and hits it SE. This may pressure Tomb into slotting Hex in to reliably threaten Bull, which forces some important sacrifice in return; Tomb may have to change up its strategy to survive or become more reliant on teammates.


No Sun? Bad news for
Ninetales and the various Chloro sweepers it supported. Without sun, Ninetales struggles to compete with the immediate power of Moltres and Magmortar. Of the Chloro sweepers, none of them have any real concrete niche, or are at least largely outclassed in any role they could take.
Others (On the Fence)
Swellow is a very strong breaker, and it likes the indirect buff to hazard removal and losing some of its most annoying answers. Granbull is an annoyance, but it's another drop that may have Swellow a bit worried: Aerodactyl. Aerodactyl has lost several key tools, but it still has access to decent Atk, good speed and STAB Stone Edge with EQ- which means that it outpaces Swellow and resists all of its attacks, even if it's not unbreakable. Swellow basically gets forced out anytime Aero hits the field, and with EdgeQuake having to hard-switch can be a bit of an issue for whichever teammate comes in. Swellow gains some benefits from the shift, but also gets a deadly new adversary.

Just to recap,
Sableye takes a small hit from having another outright good Fairy but should still be perfectly viable.

Granbull provides a very slight buff to
Primeape over other Fighting types thanks to Defiant boosting Ape should it get Intimidated. Granted, Ape still doesn't break Bull to easily, but this does pack implications for the late game, as that Atk boost can be enough to pick off Bull after it's been weakened and then force the opponent to have something that takes a +1 hit. Small, but it is a perk.
Vileplume loves almost all the new developments- no more Drought takes away its biggest issue as a defensive mon while also rendering the other Grass/Poisons largely unviable, preys on Granbull and the bulky Waters that want to handle Sharpedo and losing Arcanine means one of its worst physical matchups is out of the picture. It does still struggle with some key offensive mons, though, namely special Sharpedo, Swellow and Medicham, giving it enough issues as a wall that I'm not sure how much it really won.