I have just encounered that the AI cannot nuke you until you have nuked THEM.
AI-Civs will not (usually) initiate a first strike against a non-aggressive rival, including the human.
This is hardcoded: even if a unit has an AI-strategy other than 'Tactical Nuke' or 'ICBM' in the Editor, giving it the 'Nuclear Weapon' flag, will make the AI generally reluctant to ever use that unit in combat.
However, I have seen it happen more than once (at both Monarch and Emp) that an Ai-Civ which is losing a high-tech war (and possibly also a space-race)
will drop nukes if its situation is getting desperate (e.g. your stack of Modern Armour + Armies is getting dangerously close to its core/ capital, and you are still refusing to talk to it).
And at that point, the gloves are off. Once any player (human or AI) has dropped The Bomb, they will become a global pariah, and hence a legitimate target for any other nuclear-beweaponed AI-Civ.
Also Monarch AI cannot declare a diplomatic win even if they have the UN.
Pretty sure this is not true. But having the UN alone is not sufficient, they have to win >50% of the votes to win the election.
So if there is an even number of Civs left, say 4 Civs, the winner has to obtain 3 positive votes: 2 votes + 1 abstention would not be enough to beat the rival (who will vote for themselves).
If the UN-holding AI-Civ 'knows' it will not win, it will simply refuse to hold an election (as should the human, if they can't bribe their way to victory: see
@justanick's post above!)