Discussing time travel and those with power over time is always confusing, because things that happen one after the other chronologically are never the actual order of things. Chronologically, Nozdormu's first appearance after his creation is during the War of the Ancients. Sort of. You know, kind of. You see, the War of the Ancients trilogy (the novels) has been described as a 'time bubble' and a 'what if' scenario. So... they may or may not be canon. Nozdormu made an appearance there and played a role in things, but what he did in those books may not be what he did in the actual WoW timeline, and since time travel was involved it technically didn't happen first in the story. You follow?
Let's sum the War of the Ancients part up quick, since it might not have any actual bearing on things: Neltharion presented the Dragon Soul to his aspect buddies. Nozdormu was away being busy wrasslin' the timeline. His consort, Soridormi, stood in his place at all of the Aspect Meetings. All the Aspects juiced up the Dragon Soul with the essence of their flight, and Soridormi did that in place of her beau.
That there is the part we're not sure about. There is one part that we do know for sure, though: Nozdormu was present at the end of the War when the World Tree was planted and empowered for the Night Elves. It was through Nozdormu's blessing that the Tree gave the Night Elves immortality.
Nozdormu's participation throughout the rest of history is about the same. He buries himself in the timeline, working on this or that. Again, being an unusually busy fellow for someone who can control time. His next major appearance was at the end of the Second War. When Rhonin and his Merry Men freed Alexstrasza from the grasp of the Orcish Horde, Nozdormu arrived to save the day from Deathwing, who sought to kill Alexstrasza while she was weakened.
A few swimming-in-time sessions later, Nozdormu stuck his nose back out at the request of the Night Elves, at the end of the Third War. After the Night Elves destroyed the World Tree to save and preserve the world from the Burning Legion, Fandral Staghelm created a new tree. This tree was not only intended to replace the last tree, but to also provide a place for the Night Elves to live after so much of their territory was taken or destroyed.
Nozdormu refused to bless the new Tree, despite it being used for what the Dragons intended all along. Because Nozdormu refused, the other Aspects did, too. This didn't really please Staghelm, considering it was not the first time the Bronze Dragonflight screwed him over. After all, his son died because the Bronze wouldn't help the mortal races in the War of the Shifting Sands. Not until the Qiraji and the Silithid attacked the Caverns of Time itself, anyway.
Currently in the World of Warcraft timeline, nobody really has any idea where Nozdormu really is, except that he is 'busy' as usual. His consort, Soridormi, is once again in charge, along with his heir, Anachronos. It has been mentioned that it's unusual Nozdormu has appointed an heir (or has had one appointed for him) since Aspects are thought to be immortal as far as aging, and none have ever died... yet.
When Nozdormu was 'made' by Aman'Thul, the Titan expected Nozdormu may one day become overconfident in his abilities to control time. Because of this, Aman'Thul showed Nozdormu something a little unsettling: He showed the Timeless One that he was not so Timeless. Nozdormu has already seen when and how he will die. He knows the exact day, he knows how it will happen, and probably who will do it, too. Since the Aspects are still thought to be ageless, it's likely that Nozdormu will be done in by foul play.*
That's all we really know about the Aspect so far. He's a bit eccentric, and buries his nose in his work whenever he can, only popping up for air when he's absolutely needed. What the future holds for Nozdormu, only Nozdormu knows. Well... and Aman'Thul, I guess. But mostly Nozdormu.
There are spoilers beyond the gallery insert below. Do not read what lies beneath it if you do not want to read Wrath of the Lich King spoilers.
*That bit of information combined with the current events at the Caverns of Time has brought a pretty harsh theory to the surface: Nozdormu does not want to die, and will use the full extent of his power to prevent it. The Infinite Dragonflight are his personal agents, working to stop the death of the Bronze Aspect.
This theory has been given a little more weight from a few quests in the Dragonblight, in which you travel to the Bronze Dragon Shrine to reveal the face of the Infinite Dragonflight's leader. The face revealed to you is Nozdormu's own, though Chromie doesn't believe that's the face of the Infinite. It's just Nozdormu giving you a sign that he's still alive and kicking in his battle against the Infinite, according to her. Maybe we'll learn more in Wrath's Caverns of Time dungeon(s).