It's up to you
As a DM, you can make adjudications in a way you think would be more interesting for the group. Consider following advice from the AL Dungeon Master's Guide:
Always follow this golden rule when you DM for a group: Make decisions and adjudications that enhance the fun of the adventure when possible.
The "Instant Death" rule is intended for using on player characters, not monsters, since only player characters normally gets death saving throws. As a DM, you might use the "Instant Death" rule on monsters as well, if the monster has its own ways to avoid death on 0 hp (apparently, Undead Fortitude does exactly that). It is perfectly fine — many DMs use PHB rules to adjudicate monster's actions (for example, they can say Grapple special attack can be made by a monster).
Zombie probably die from massive damage. The rule says:
Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.
The rule explicitly says "you die", not "your hit points becomes 0", thus, does not trigger the Undead Fortitude feature. The Undead Fortitude feature explicitly says it works only when damage drops the undead to 0 hp:
Undead Fortitude. If damage reduces the zombie to 0 hit points, it must make a Constitution saving throw with a DC of 5 + the damage taken, unless the damage is radiant or from a critical hit. On a success, the zombie drops to 1 hit point instead.
On the other hand, as @JohnMontgomery has mentioned, Undead Fortitude says it drops to 1 hit point "instead". So if it saves then technically the zombie is never reduced to 0 HP and massive damage can't trigger.