I think shortly after the main quest, Nerevar took a short vacation to Cyrodil to celebrate with his good buddy Caius Cosades. Somebody overheard them talking about the "expedition to Akavir" they were going to take in Caius' smoking room that weekend, thus starting the rumor we hear in Oblivion.

"Vivec's Nerevar's lover, Almalexia's Nerevar's lover, Dumac's Nerevar's lover, I'M Nerevar's lover! Are there any other Nerevar's lovers I should know about?" (a Hist tree slides into frame)

1:43 dagoth ur definitely still lives.  I just saw him in a prisoner cart with Ulfric Stormcloak and the Dragonborn.

Dumac will be really confused when an Argonian with a fur hat tries to rescue him

A dwemer fact that gets overlooked is that they were in possession of and Actively studying the Dragon Elder Scroll when they vanished. The very same Scroll that sent Alduin into the 4th era.

Nerevar, faravar, whereverar you are-avar

The Nerevarine is an argonian. So, they returned to black marsh and was why the oblivion crisis was easily held back.

I don't know where THE nerevarine went, but I'm willing to bet MY nerevarine vanished after her signature "Mad Hops" spell sent her flying straight across Vvardenfell. Hell, she could have accidentally landed in Akavir. Those were some powerful jumps. (I desperately miss jump & fortify acrobatics lol)

Dwemer are time traveling Dunmer, think about it or don't, actually don't

I think it is important to note that Dumac is still at fault for the battle of red mountain.
He was asked multiple times by one of his closest friends "hey, are you whorshipping Lorkhan's heart and building a brass god?"
And instead of thinking "maybe I should investigate this, I don't think he'd just make this up" he acted all insulted.
He is a king, and as such us responsible for the actions of his people.

The Nerevarine obviously settled down with ahnassi and spread the Akavir rumor so people would stop bothering him

The original villain of Skyrim was supposed to be the lost Dragon Emperor from the disaster of Ionith. I personally think all the foreshadowing of Akavir was leading to this and the Nerevarine might've been mentioned more or even appeared in Skyrim.

The letters between two friends bit is great

I hate that the best conclusion Bethesda could think of for the Nerevarine was "Let's just send them to Akavir and make the fans wonder about what it means for decades."

:For so long we've been fixated on where the Dwarves disappeared to, that I think we've forgotten to ask: from whence they came?"

In ESO there's a book called Aurbic Enigma 4: The Elden Tree (we never actually get to read the first three Aurbic Enigmas), which has a very interesting passage about the origin of elves.

"The spike of Ada-Mantia, and its Zero Stone, dictated the structure of reality in its Aurbic vicinity, defining for the Earth Bones their story or nature within the unfolding of the Dragon's (timebound) Tale.  The Aldmeri or Merethic Elves were singular of purpose only for so long as it took them to realize that other Towers, with their own Stones, could tell different stories, each following rules inscribed by Variorum Architects.  And so the Mer self-refracted, each to their own creation, the Chimer following Red-Heart, the Bosmer burgeoning Green Sap, the Altmer erecting Crystal-Like-Law, et alia."

Essentially, by imitating the creation of the Adamantine Tower, using which the gods ended the Ehlnofey wars and imposed linear time on Nirn, the Aldmer could impose their own will on the world around them, which in turn would shape them into the different races of elves we find in game.  The rest of the text mostly concerns itself with the White-Gold Tower, through which the Ayleids became the Ayleids, and Green-Sap, a massive extra-temporal tree grown by the Bosmer following the teachings of Y'ffre (though the in-universe author, being a Direnni High Elf, uses the name Jephre).  There's also stuff there providing background context for the story of the area where the book is found in ESO, but that's tangential to the subject at hand.

The in-universe author, Beraldalmo the Signifier, could be partially or wholly mistaken, that's just how TES lore is sometimes.  But if we follow his line of thinking, it leads us to an interesting conclusion regarding the Dwemer.  The Three Good Daedra taught the prophet Veloth, who led the Chimer to align themselves with Red-Heart—the Red Mountain, with the Heart of Lorkhan as its Stone.  Other sources in ESO point to a very clear Tower for the Dwemer—Walk-Brass, the Numidium, with the Heart of Lorkhan as its Stone.

Notably, the creation and activation of Numidium was actually the last thing the Dwemer did before their disappearance, not the first thing they did as a distinct people.  The Numidium is also known to pretty much break time whenever it's turned on, so being involved in some manner of retro-causal shenanigans is honestly par for the course.  But it would also mean that the Dwemer both began and ended at the same point, in the Red Moment.

If we take that train of thought, and pair it with the idea that the Nerevarine wants to Make Things Right with Dumac and the Dwemer, that's the lead to be investigated.  Of course, Numidium was destroyed in the Warp in the West, and the Nerevarine was personally responsible for making the Heart of Lorkhan disappear.  So what's a reincarnated ancestral culture hero to do?

A few options come to mind for me.  Maybe Sotha Sil had some useful notes squirreled away somewhere in the Clockwork City. Maybe they could find some way to access the Mantellan Crux like the protagonist of Daggerfall.  Maybe the Heart Crystals we find in Skyrim's Dragonborn expansion would resonate with it in some way and could be used to track it (though they would have been inside Red Mountain at the time, not on Solstheim).  Honestly, my first instinct in that situation would be to talk to Divayth Fyr and Yagrum Bagarn, to see if I couldn't work out some ideas with them.

It doesn't really come to a single answer for where the Nerevarine may have gone, ad it does lean on a fair few assumptions.  But it's an interesting line of speculation, I think.

My degree was in linguistics and I have a lifelong interest in language reconstruction. I recently took up a project where I catalogged all of the attested Meric languages' words and grammar to see if they were made carefully enough to reconstruct some Proto-Mer. They weren't, however, if we're to take Dareloth's approach of treating these like actual historical artifacts, I found it interesting that the Dwemer language is the most divergent branch, considering that the Dwemer appear to have been as long-lived as most Mer, and that their turnover rate for new speakers (one of the driving forces of sound changes) would have probably been quite low. This would suggest a much older linguistic split, as it seems like it was mutually unintelligible from contemporary Meric languages as of the sequestering of the Falmer. My guess would be that they parted ways from other Mer shortly after the collapse of Old Ehlnofey.

I like the idea but am not inclined to theorize a new landmass with no evidence, given that we already have a list of existing islands off the eastern coast.
We know the Nerevarine came into direct contact with Yagrum Bagarn, the last living dwemer, and Yagrum avoided the disappearance by being "in an outer realm" instead of nirn. There seems to be quite a lot of mortals that live in planes of oblivion, as seen in Shivering Isles and a load of ESO expansions, so it seems to me the simplest and most direct way in which the Nerevarine would choose to go look for surviving dwemer would be to go dimension hopping through the planes in search of a group of surviving scholars.

Even if we remove the "he's trying to find the dwemer" idea I think that given his standing as a prophesized champion of Azura who is tasked with shepherding her people, but has noticeably vanished off morrowind itself, I'm inclined to think he's likely in Moonshadow.

Perhaps the Ehlnofey who became the Dwemer were stranded on a hostile and mountainous island. They survived through ruthlessness and cunning, denouncing the gods whom they believed abandoned them on this blasted rock. Over time these traits crystallized, and while memory of the mountain island may have been lost to the sands of time, it’s impact on the ancient Dwemer could be seen in the eras that followed

The Nerevarine is a massive loot goblin and goes where the magic treasure is.

I get what you’re saying but at the same time this requires a lot more headcanon than the Nerevarine going to Akavir to see if there’s a threat of another invasion.