Okay, it's hard to be certain for the Dragonborn because, ultimately, his/her history has not been written yet.
But, if we look at The Champion of Cyrodiil, there is fairly strong evidence that they became Sheogorath. The hints of this are in Sheogorath's dialog in Skyrim itself.
From this reddit thread: https://elderscrolls.fandom.com/wiki/Thread:398774
The dialog and related quests in ES4:
In skyrim he says "You know,I was there for that whole sordid affair.Marvelous time! Butterflies,blood, a Fox,a severed head...Oh,and the cheese! To die for.
U see butterflies-When u enter shivering isles whole room turn in butterflies
Blood-Blood of Aedra and Deadra for Martin's quest.
Fox-Gray Fox from thiefs guild
Severed head-head of Mathieu Bellamont's mother in Dark brotherhood quest.
Cheese-Quest for Sheogorath in Oblivion,where u must steal all cheese from khajiit's.
If we assume that's true, and we also assume that The Champion of Cyrodiil also performed tasks for the other Daedric Lords, then it shows it's possible for one of the Lords to "win". Now, admittedly, this is a lot of speculation and assumption, but there are hints.
So, what does this mean for the Dragonborn? It means that one of the aspects will "win" over the others. For YOUR character, I think it's reasonable to pick who that is... but, we're going to have to wait until ES6 to hear/read about what happened to the canonical Dragonborn.
Now, in my opinion, the answer is: The Dragonborn end up in Sovngarde... even if you're not a Nord. There was some dialog there that hints that way. (Emphasis below is mine)
Tsun: "That was a mighty deed! The doom of Alduin encompassed at last, and cleansed is Sovngarde of his evil snare. They will sing of this battle in Shor's hall forever. But your fate lies elsewhere. When you have completed your count of days, I may welcome you again, with glad friendship, and bid you join the blessed feasting."