Once you announce that you’re casting Mutual Destruction, players can’t try to remove your permanents with flash to make it lose flash until you’re done casting it. If it loses flash after it’s been cast, it will still resolve if able.

You must sacrifice exactly one creature to cast this spell; you cannot cast it without sacrificing a creature, and you cannot sacrifice additional creatures.

The last time that the game checks whether you have permission to cast a spell is before costs are paid. If you sacrifice a creature with flash to cast Mutual Destruction and then have no other permanents with flash, you’ll legally finish casting Mutual Destruction even though it no longer has flash. In this case, abilities that trigger whenever you cast a spell with flash won’t trigger.