Like all effects that are worded "until end of turn", the Nexus' animation ability wears off during the cleanup step, which is the second step of the end phase, after the end step. There is no way to make an effect that ends during the cleanup step carry over to the next turn.
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I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
It turns back at the beginning of the cleanup step. The cleanup step is the step after the end step, in which players discard down to 7, damage is removed from all permanents, and "this turn" and "until end of turn" effects end. Normally, players don't get priority during the cleanup step, but it can happen (such as with Madness cards) - if that ever happens, the cleanup step will just restart again and again until nothing happens during it.
In short, it's utterly impossible to activate the Nexus in one turn and have that activation carry over to the next. "Until end of turn" and "this turn" effects are smarter than that.
Maybe this reality is actually an alternate reality of a reality in which New Phyrexia was the joke expansion and Mirrodin Pure was the obvious expansion, except our reality is the Time Shifted reality which is opposite of that other reality which is that actual reality.
-Cards where you can "cheat" the end step in the way you are describing are worded "At the beginning of the next end step"
-Cards where you cannot "cheat" the end step in this way are worded "until end of turn" or similar
Just a hypothetical..
What if one were to throw the inkmoth nexus into a Tawnos's Coffin before the clean up step, then let it out during the next untap step?
Would that allow it to remain animated through turns?
Nope, each time it leaves and comes back it becomes a new object and forgets everything about it's previous existence.
So it'll come back as a unanimated land that will have summoning sickness, if animated.
Just a hypothetical..
What if one were to throw the inkmoth nexus into a Tawnos's Coffin before the clean up step, then let it out during the next untap step?
Would that allow it to remain animated through turns?
No.
400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence.
There are exceptions, but this isn't any of them. Even phasing, which works the way a lot of people want exiling and returning to work, won't let you do this.
Maybe this reality is actually an alternate reality of a reality in which New Phyrexia was the joke expansion and Mirrodin Pure was the obvious expansion, except our reality is the Time Shifted reality which is opposite of that other reality which is that actual reality.
My opponent's end step - I activate Inkmoth.
Does inkmoth stay a creature during my turn?
Does it turn back at the beginning of the end step, or at the end?
In short, it's utterly impossible to activate the Nexus in one turn and have that activation carry over to the next. "Until end of turn" and "this turn" effects are smarter than that.
Level 1 Judge. I tweet. Member of clan <Limited>. The Dunning-Kruger effect.
-Cards where you can "cheat" the end step in the way you are describing are worded "At the beginning of the next end step"
-Cards where you cannot "cheat" the end step in this way are worded "until end of turn" or similar
L2 Judge
What if one were to throw the inkmoth nexus into a Tawnos's Coffin before the clean up step, then let it out during the next untap step?
Would that allow it to remain animated through turns?
So it'll come back as a unanimated land that will have summoning sickness, if animated.
No.
There are exceptions, but this isn't any of them. Even phasing, which works the way a lot of people want exiling and returning to work, won't let you do this.
Level 1 Judge. I tweet. Member of clan <Limited>. The Dunning-Kruger effect.