Let's go broad strokes here, to get you pointed in a direction. You want to represent the five apocalypses, so the first thing you need is something that captures the identity of each. So what have we got to work with?
- Divine (WB)
- Mechanical (UR)
- Zombie (BG)
- Nuclear (RW)
- Alien (GU)
To make a diverse set, you'll probably want each to have a couple of core creature types, and a signature mechanic. So let's start by breaking down the identity of each.
Divine
The obvious choice here would be to make the core types angels and demons. However, let's look at the description you wrote;
"On the Day of Ruin, the skies themselves ripped open, and grotesque, but beautiful creatures fell from the heavens, slaying all in their path and chanting about how we were all nonbelievers. Nowadays, many have been enthralled by these so called "Angels" into worshipping them. But who sent them and why?" Now, there is the classic description of angels as terrible and unsettling to behold, and Angel + Horror would definitely be an option here, but there's another interesting option that presents itself here. The way you've written this implies that while people
believe that they're angels, that's not what they truly are. So what about a different focus? Perhaps the "angels" are simply monsters, or Eldrazi. For a second creature type, rather than the beings themselves, focus on their cultists. The obvious choice for a second creature type would be Cleric, but you could also do something like bringing back the obsolete Heretic type or even creating a Cultist type, if you really wanted to emphasise the difference between them and normal religious people. Or go with Minion for a less bold but still unique choice.
For a mechanic, think about what these beings do. Their main focus seems to be on destroying what they see as blasphemous, and on converting others to their worship. While either of these would make a solid theme to base a mechanic on, just having powerful creatures can cover the first (especially if you go the Eldrazi route, and include some of their associated mechanics). So think about how you might mechanically represent the process of religious conversion, keeping in mind that you want your signature mechanic to be something that can be displayed at at least Uncommon, and ideally at Common. Which means that big, swing-y mechanics like permanent stealing aren't a good fit, despite how obvious they are, as these aren't fun to play against if they happen often (and usually have to be aggressively costed to feel fair). My recommendation is "Awe Counters" to represent non-believers being influenced gradually to be more and more cowed by these divine beings. Some cards might simply spread the counters around, others might remove them to get an effect, and still others might be enabled by permanents (most likely creatures) having a certain number of counters. There might or might not be a way for the player to remove them. By making the core element an enabler, rather than a direct effect, you get a wide design space to play in, while still keeping the same unified theme of spreading influence.
Mechanical
I think it goes without saying that the backbone of the Mechanical creatures will be Constructs. That doesn't need to be questioned. However, there are some interesting options for other creature types. For instance, Illusions could represent holographic projections. Assembly-workers make a solid choice for re-purposed industrial robots, but you could also play fast and loose and use the normally Eldrazi-only Processor type for everything from automated garbage trucks to horrific clouds of nanomachines (which could also be Oozes). However, not all machines are robots, and there's an interesting option there. Your blurb on the mechanical apocalypse mentions a rogue A.I. One way to represent artificial intelligences would be to give enchantment creatures exclusively to the mechanical faction, which might in turn tend itself to them having an enchantment-matters theme for their core mechanic.
Alternatively, you could crib from the Phyrexians, who also sought to replace biology with technology. You could go the safe and boring choice of just being the most artifact-friendly faction. Representing the construction of mechanical armies through token generation is also a solid choice. The important thing here is to understand first
why the machines want to purge organic life, and second
how they intend to do it, then make a mechanic that encourages a play style representative of those intentions.
Zombie
This is another one that seems like it would be a monotype. However, monotypes are boring. We want options. I recommended Horrors for the "angels" before, but they'd fit in just as well here to represent twisted abominations of flesh beyond just mere zombies (this is a good time to mention that Horror is actually a good fit for
all the factions for different reasons, and if you didn't mechanically tie it to one in particular, having them in all five is a totally valid way to show how terrible the apocalypse can be). Mutants could work for that as well, but I think they fit better somewhere else. Vampires could appear here to, as a
less severe form of the infection. Of course, having Green in the mix means that Beast is a great option for infected animals, but those could just as easily be
Zombie Beasts, which means they wouldn't really be a separate type. However, we could have abominable undead amalgamations of creatures as Chimera. Ooze could also fit here, for things so far gone that they're basically just a shambling mass of rotting flesh and sickness. There's another interesting option, though. If you were to replace "experimental virus" with "experimental
bioweapon" then we could also fit Fungus or Insect in, as possible alternative sources for the infection.
Mechanically, bringing back Infect or some other token mechanic similar to what I suggested for the Divine faction could work (though for some players it can be confusing to mix token types). However, it's good to consider other options. Graveyard retrieval is obvious. Perhaps using something like Regeneration or Indestructible to show that the undead will just keep getting up unless destroyed utterly. In that case, it's important to remember that while Indestructible has a lower skill ceiling to understand, it's less fair than Regenerate - which has a built-in "dial" to adjust its impact - at low rarities. My personal thought on the "just won't die" angle would be to make a new mechanic that serves as a "fixed" version of Regenerate.
Example: Reanimate N (When this creature dies, you may pay N and return it from your graveyard to the battlefield tapped. Effects which trigger when a creature enters the battlefield are not triggered.)
Another option is either reanimating creatures you kill or getting token generation when a creature dies, representing the creation of new zombies, but this can be difficult to do in a way that's fun for all players at Common. You would probably need to include an associated cost or drawback, or a way for the defending player to prevent it. Lastly, you could find a way to represent the classic "zombie horde", such as taking inspiration from (or even using) Banding or Amass.
Nuclear
This one's an interesting one. I think that for a creature type, Mutant is the first pick. How better to represent a world turned into a radioactive wasteland than to show how it's twisted those within it? Giant insects could work, and they are seen in Red as well as Green, so they're not totally off-colour. Big bugs could still be
Mutant Insects, though. What about Soldiers or Rebels to represent the remains of the armies that brought the world to destruction, and those that fight against them? Or, since you specified
aetheric weapons, rather than nuclear in the traditional sense, we could have Aetherborn here. They were Black-aligned previously, but there's nothing strictly tying them there, especially if they were made by other means. ...though in truth, your description makes me think you thought of these already.
For a signature mechanic, something encompassing the concept of radiation seems good. Since there's so many other good places for a token-based mechanic, I'd avoid that here. Instead, let's look at mechanics like Flanking, Bushido, and Afflict, which punish blocking. Poison counters could also be fair game, as putting counters on the player doesn't interfere with creature-counter bookkeeping. It doesn't necessarily have to be poison counters themselves, either. The Nuclear family (couldn't resist) could just as easily use some kind of unique counter here with its own penalties for accumulating, not necessarily a just a win condition. Perhaps also a way to get rid of them? The fact that poison counters are permanent is one of their key flaws, in my opinion, since it means they have to be used more sparingly in a set (I should note that WotC R&D disagrees with me on this point). If you did go this route, I'd suggest a one-two punch. Lingering counters for the player, and non-lingering "radiation sickness" for blocking creatures, to set up a painful choice. Consider making Nuclear's mechanic with a self-destructive aspect that encourages risky play, and hurting themselves in the long-term for immediate advantage. Also, maybe talk to
spazlaz about the radioactive mana they came up with.
Alien
You might or might not have known, but WotC has actually shown off an Alien creature type on one of their test cards. Of course, any fantasy race could be alien in a world that doesn't have them, and there's nothing stopping you from having space elves or space orcs either. Or, for that matter, virtually any species that is strange looking. Maybe you could have a race of Slug aliens, with Spikes as their warrior caste. Shapeshifters are weird enough to be alien, whether the creatures themselves, or their creations. Drones work here for a secondary type (I didn't mention Drones for the Mechanical faction, but obviously they work there too). Or perhaps you could bring back the Ship type in defiance of the convention that their type is defined by their crew, to represent the strange crafts in the sky. Maybe have the second type be Plants and emulate the invasion of the pod people, use Homunculus for artificial servants, or even Weirds. It's hard to give any suggestions for what creature types work well with aliens, since most things
could be an alien. Obviously, things like Pilots will work well, Artificers can fill the role of scientists, and Soldiers go well with an invading force.
For similar reasons, it's hard to recommend anything mechanically for a group whose intentions are unknown. You touched on abduction and disintegration, which are both good concepts to base mechanics on, but perhaps not good to use as a central mechanic. Aliens are another place where being artifact-friendly fits, though. Vehicles in particular would be something they'd reasonably use a lot. Other than that, the only good alien-related word that comes to mind that might make a good inspiration for a mechanic is "probe".