I have an Arcanist Owl in play and Thassa, Deep Dwelling for a total of 5 devotion to blue.
Thassa is a creature. I mutate Illuna, Apex of Wishes onto Thassa with Thassa on top and Illuna on the bottom.
After this resolves my opponent kills my Arcanist Owl reducing my devotion to blue to 1. Thassa becomes an enchantment again.
What happens to the mutated Illuna? What happens to Thassa? What happens if I put Illuna on top and Thassa on the bottom?
The exact rules for the mutate mechanic haven't been released yet.
According to the mechanics article for Ikoria: Lair of Behemoths, a mutating creature spell (e.g., Illuna here) doesn't enter the battlefield; rather, it merely changes a permanent's characteristics (e.g., those of Thassa here) when it resolves. But nothing in the mechanics article suggests that these changes are active only while the permanent is otherwise a creature.
A mutating creature spell doesn't enter the battlefield as a separate permanent; rather, it becomes part of the permanent it targets, changing that permanent's characteristics. But nothing in the comprehensive rules suggests that these changes are active only while the permanent is otherwise a creature.
See also this thread. That article, however, also leaves open what the exact nature of these changes is; see also this thread. Accordingly, the answer to your question is not yet known, pending an update to the comprehensive rules.
EDIT: Struck out certain text after comment 3 was posted.
EDIT (May 19): Edited in view of rule update.
We already have the answer to this thanks to the AMA on Reddit. Here is the quote:
Q: How does mutation effect devotion to the Theros gods? Also, if a mutation is on the God and if devotion turns off for whatever reason, does it fall off?
A: Nope; mutating doesn't create an attachment, it's just merging two cards into one. The resulting permanent just stops being a creature. This means, for example, that if you merge a creature on top of Thassa, you can get a permanent with no card types!
Since the creature has a whole now has the ability to turn itself "off" it will do so just like the Gods do. In your scenario, the monstrosity you have is not a creature until you get back to 5 Devotion. It doesn't matter what is on top or the bottom but they do remain "stuck together".
The original question asked, what happens if my Mutate target ceases to be a creature before the mutate spell resolves, is also answered in themechanics article:
if a mutating creature spell tries to resolve but can't because its target is illegal or has gone missing, the mutating creature spell will simply resolve and enter the battlefield.
The mutate spell has a target of a non-human creature you control. If you target Thassa and she stops being a creature before the spell resolves, your mutate spell enters as a creature, instead of mutating.
Thassa is a creature. I mutate Illuna, Apex of Wishes onto Thassa with Thassa on top and Illuna on the bottom.
After this resolves my opponent kills my Arcanist Owl reducing my devotion to blue to 1. Thassa becomes an enchantment again.
What happens to the mutated Illuna? What happens to Thassa? What happens if I put Illuna on top and Thassa on the bottom?
Cards:
https://scryfall.com/card/thb/71/thassa-deep-dwelling
https://scryfall.com/card/iko/190/illuna-apex-of-wishes
https://scryfall.com/card/eld/206/arcanists-owl
I know that equipment and enchantment creatures auras fall of, but do mutated creatures too?
The exact rules for the mutate mechanic haven't been released yet.
According to the mechanics article for Ikoria: Lair of Behemoths, a mutating creature spell (e.g., Illuna here) doesn't enter the battlefield; rather, it merely changes a permanent's characteristics (e.g., those of Thassa here) when it resolves. But nothing in the mechanics article suggests that these changes are active only while the permanent is otherwise a creature.
A mutating creature spell doesn't enter the battlefield as a separate permanent; rather, it becomes part of the permanent it targets, changing that permanent's characteristics. But nothing in the comprehensive rules suggests that these changes are active only while the permanent is otherwise a creature.
See also this thread.
That article, however, also leaves open what the exact nature of these changes is;see also this thread.Accordingly, the answer to your question is not yet known, pending an update to the comprehensive rules.EDIT: Struck out certain text after comment 3 was posted.
EDIT (May 19): Edited in view of rule update.
Since the creature has a whole now has the ability to turn itself "off" it will do so just like the Gods do. In your scenario, the monstrosity you have is not a creature until you get back to 5 Devotion. It doesn't matter what is on top or the bottom but they do remain "stuck together".
The mutate spell has a target of a non-human creature you control. If you target Thassa and she stops being a creature before the spell resolves, your mutate spell enters as a creature, instead of mutating.
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