I believe the Nurgling would be dead first. Mutation is a forced interrupt and Banner is a reaction. They both would be triggering off of step 1 of the attack process "declare the attacker," but from the language on the cards it seems pretty clear to me that DM would trigger first, "when attached unit exhausts" vs "after attached unit is declared as an attacker." I think in general that interrupts are resolved before reactions if they are triggering off the same thing.
If you shield the DM damage to keep the nurgling alive I dont think you would be dealing 4 damage though. As far as I can tell it would just be 1 moved due to the banner, and the normal 2 for the attack.
Part of me wishes that Dire Mutation wasnt restricted to army units as slapping it on Ku'gath with the banner seems like it could be a powerful combo if I understand the timing correctly. You would lose the ability to heal 2 hp per attack, but you would still be able to heal 1hp per attack and you would guarantee that he always had at least 1 damage to move. But I understand why it's army only as putting up to 3 on an enemy warlord would be too nasty.
Interrupts occur before the triggering condition completes. They actually "interrupt" the triggering condition to modify the state, then the triggering condition continues, then reactions may trigger in response to the triggering condition completing.
Per the RRG: Framework Event 184.108.40.206 The attacking player chooses one ready unit he controls at the planet, and exhausts it to attack. This declares the chosen unit as the "attacker". Exhausting the unit in this case is the same as declaring it as an attacker (note that Dire Mutation would also trigger due to other exhaust reasons as well, hence the different language). Since Dire Mutation interrupts the act of exhaustion, the unit is killed before it finishes exhausting. Since it never exhausts, it is never declared as an attacker, and therefore the banner does not go off (and the attack doesn't happen).
I believe this is the correct reading of the rules, though this probably belongs in the rules forum..