Alright lets takle this cause' no one else knows their stuff. (Although that's just because I've played all the games and read all the WarCraft books, sort of gives me an edge.)
Lore nerds, I'm in dire need of your counsel. I'm making a campaign unveiling an alternative timeline where the Burning Legion won the battle of Mount Hyjal.
Well, that's not good...
What would happen in this case?
Alright so this kind of depends if you want to follow the lore of War of the Ancients or Chronicles. The lore of WotA states that the Legion was summoned to Azeroth by the Old Gods so that they could escape from their prisons, to do this however Sargeras would have to be summoned. The reason for this is as Korialstrasz says, the free races of Azeroth would always have hope even if all demons save Sargeras would step onto Azeroth, but should Sargeras step onto Azeroth there would be no hope to stop the utter destruction of Azeroth as we know it. So Archimonde would unquestionably use his powers to summon Sargeras unto Azeroth.
Now we need to talk about the Old Gods again, because Sargeras is being attracted to Azeroth by the Old Gods because as Korialstrasz puts it, "if the old gods were to step onto Azeroth in its true form even Sargeras would be brought to kneel." We know from WotA that there could only be two sources for this information, the titans (who personally knew Sargeras or Nozdormu who can travel in time.(It can't really be anyone else as Sargeras is not widely known about, not even Azshara (the most powerful mortal) knew of Sargeras before he revealed himself.)) So regardless of which of the two sources revealed this information (or maybe it's both) the source is very reliable. Keep in mind, Korialstrasz is not the only character we see frightened by the Old Gods, Alexstrasza is significantly more frightened by the Old Gods and Nozdormu says that the Old Gods are "far more ancient and malevolent than Sargeras." So if you go down the WotA route then Archimonde would summon Sargeras which would cause the Old Gods to break out of their prison and rip the legion to shreds. This is really useful if you want the mortal races to make a last stand before finally perishing. (Wyrmrest Accord would be the natural location for such a last stand.)
Now for the lore as it is depicted in Chronicles... Chronicles is an absolute cluster f--k, the lore in Legion states the same thing, the Old Gods tried to summon Sargeras but with one key detail, in Chronicles Sargeras could poke the Old Gods to death... and the Old Gods know this as they witnessed Y'Shaarj, the most powerful of the Old Gods being ripped straight out of the ground by a Titan. So why did they want to summon Sargeras to Azeroth? To loose... it really doesn't make sense, they knew Sargeras would rip them to shreds but tried to summon him anyway... This book is basically why I hate WarCraft lore now... Don't get me wrong I still follow it, intensily, but it has dropped significantly in quality. Let me put it in perspective, Warlords of Draenor made more sense than Chronicles, the ending of the Cataclysm made more sense than Chronicles. So please as someone who loves the WarCraft III lore I beg you, go with the WotA version.
He'd destroy Nordrassil and gain access to unfathomable powers granted by the Second Well of Eternity. Given how he initially opted to serve under Sargeras out of hunger for power, would he truly remain loyal to Sargeras and pursue the invasion of Burning Legion?
If Warlords of Draenor is anything to go by he probably would, Archimonde summoned the mortals fighting him to the Twisting Nether (risking his own life) to fulfill the ambitions of Sargeras. Besides if Archimonde ever died he would get teleported straight back into the Twisting Nether (the domain of Sargeras), which would be bad. I think Archimonde is too clever to betray Sargeras. Especially considering that War of the Ancients goes into detail on what Sargeras' torture is like, and I would to put it mildly not want to be tortured by Sargeras. In Legion it also seems as though Kil'Jaeden and Archimonde were convinced by Sargeras that the Legion was doing the greater good so it's quite plausible that Archimonde believes in the Legions cause.
In short, yes he would most likely stay loyal.
If he does, could Azeroth's denizens be capable of opposing him any further?
Once again I'd defer to Korialstrasz who said that just as long as Sargeras doesn't step foot on Azeroth there's hope. But keep in mind that if you follow the lore of Chronicles then War of the Ancients is non-cannon.
I wouldn't mind the campaign ending in such an apocalypse by the way,
If it does end in an apocalypse it would once again make more sense if the foe were the Old Gods.
I'd just need a rough summary on how such scenario would be carried out in that case: would Archimonde be able to summon Sargeras on the spot or would he have to hound some magical relics?
He could summon him on the spot although casting such a spell would undeniably take time as accourding to War of the Ancients it took the combined power of all dragons (in the form of the dragon soul) quite a while to pull of.
Would Sargeras then slay the nascent Azeroth, who isn't yet a titan and hence defenseless? In what space would this happen, surface of Azeroth?
It wouldn't really happen at all if you go with War of the Ancients but isntead the Old Gods would corrupt Azeroth until it became the puppet of the Void Lords (the masters of the Old Gods.) But if you want to go with Chronicles then the story is vague, contradictory and non-sensical enough for you to go down any route you want.
If you have any questions please ask and I'll answer to the full extent of my knowledge.