It makes the zombies more effective against weaker foes, but won't break the game
Even with the zombies having a d12 damage dice instead of a d6, there are plenty of other spells that deal more direct damage. Spirit guardians can deal 3d8 damage over and over again to a larger number of enemies for up to 10 minutes. Heck, the level 1 spell Burning Hands does 3d6 damage in a small cone, which may end up dealing more damage than the 1 zombie. The axe adds, on average, an extra 3 damage which is really only significant against weak creatures with low HP that the zombie can easily hit.
Undead Fortitude isn't really a concern either
You mentioned a concern that Undead Fortitude would make the zombies more effective, leading to an overall unfair boost in damage. But remember that the DC for the save is 5 + The damage taken. As the players fight stronger opponents, the average damage they will encoutner will rise, and the saving DC will grow quickly. For some effects it will become impossible for the zombie to save (Unless you play with critical success saving throws).
For example, Animate Dead can be cast starting from a 5th level caster. At this level, The CR 5 Roper deals on average 22 damage from its bite, giving the zombie a DC 27 save. Given the Zombie's bonus of +3 to Con Saves, this is impossible for the zombie to make alone. A Hill Giant club attack will do on average 18 damage; a DC 23 Save for the zombie. It will only pass with a 20 on the saving throw. A creature only has to deal 8 damage to a zombie for the saving throw to be a coin toss. Zombies have an average of 22 hp, so any creature, like the roper, which can knock it down in one hit is pretty much guaranteed to keep the zombie dead, even with undead fortitude, unless the party uses a questionable amount of resources to bolster its save.
Even if the zombie lasts a while, the damage bump is minimal
Even if the zombie is lucky and gets in 3 hits before it dies, that's on average, a difference of 9 damage. CR 5 monsters hover around 100 hp, so over 3 attacks, they are down an extra tenth of their health. Not that big a deal. Plus This is around the point where the players will face creatures that have resistances/immunities to non-magical damage, or that can regenerate hp. So, unless the players really deck out their zombies, each one may only be half as effective as usual.
For those reasons, I would let the players have their fun and allow the axe zombies.