These are the Ork forces that were identified as participating in the Third War for Armageddon in Codex: Armageddon (3rd edition):
These are the forces of the Imperium that were identified as participating in the Third War for Armageddon in Codex: Armageddon (3rd edition):
In the codex, Sons of Guilliman is spelled "Sons of Guilleman."
At that time, the Astra Militarum was known by its Low Gothic name: Imperial Guard.
At the time that the codex was published, the Deathwatch had not been introduced yet. They were named later when the Inquisitor game was published. Conventional wisdom is that the "Kill-teams" of the Ordo Xenos refers to the Deathwatch. That's the theory to which I subscribe. However, it could be that these "Kill-teams" may have been Inquisitorial Storm Troopers (akin to Tempestus Scions), so I haven't realigned this listing to the Adeptus Astartes column.
The Imperial Knights are not listed, but it seems likely that many Imperial Knight Households sent detachments and more than a few Freeblades were probably present. In addition, while the Imperium-aligned Knight Households might be listed separately, the Adeptus Mechanicus-aligned Households might be listed under the Adeptus Mechanicus heading. Since none were listed in the codex, however, none are shown here. It would be very interesting to see an update of this campaign with Imperial Knights listed.
I haven't renamed the company equivalents of the Adepta Sororitas to their current nomenclatures (presumably these are Commanderies).
While the Adeptus Arbites are listed, the Enforcers of Armageddon (or whatever Armageddon calls the local constabulary) aren't. There should be at least as many precincts of these law enforcers as their Adeptus Ministorum counterparts, and more likely many more.
The Adeptus Mechanicus has been fleshed out in the years since 3rd edition. I would have listed the Titan Legions under a sub-heading of "Collegio Titanica" but I wanted to preserve the basic structure of the official chart. Similarly, I don't know if the "regiments" of Skitarii would be Cohorts or Maniples. In addition, the Legio Cybernetica has been brought back.
The Officio Sabatorum and Templars Psykologis remain enigmatic, their roles not being officially revealed in any official publications that I can think of (admittedly, I haven't read the novels). We might speculate on their natures based on their names and listings (e.g., the agents of the Officio Sabatorum might be similar to the Assassins, but adept at sabotage instead of assassination; while the Templars Psykologis work in squad-sized teams and probably have something to do with either psychological warfare or psyker warfare).
I imagine that an official updated version of this campaign would expand the listing of the forces available.
When the [Third War for] Armageddon campaign took place, hobbyists could play battles using regular 3rd edition Warhammer 40,000, as well as Epic Armageddon (playing larger battles, though this game no longer exists and the closest thing we have now is Apocalypse) and Battlefleet Gothic. You can find the rules for both of these games online (see the Resource topics in the Other Games forum). Aeronautica Imperialis is a great way to cover the aerospace battle (between the space battles of Battlefleet Gothic and the land battles of Warhammer 40,000). Shadow War: Armageddon (no longer supported) and Kill Team provide great ways to play the battles at the squad/warband skirmish level. If GW ever gives us good rules for playing Adeptus Titanicus in the "modern" age (or if enterprising hobbyists develop some rules ), players can play the Gargants versus Titans/Knights level.
Another great opportunity for a project is Necromunda. In The Battle for Armageddon, there's a page with a transcription of a communication to a Commissar in which a Captain (of some Armageddon military/law enforcement forces) identifies a number of gangs that have agreed to cooperate (i.e., fight alongside/in support of military/law enforcement forces). It would be interesting to see an enterprising hobbyist develop rules for Ork "gangs" in both the hives as well as the wastes so that players could represent this facet of the war in the Necromunda game.