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I have been thinking over this and couldn't find anything about it online so far.

If I animate a corpse into a zombie to be my undead follower using the spell animate dead, when it is reduced to 0 HP will it get the benefit of its Undead Fortitude trait? If so, if it is then not slain, will I still retain the control over it afterwards?

The zombie's Undead Fortitude trait says:

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. (Basic Rules, p. 161)

An animated zombie benefits from Undead Fortitude as normal

The feature Undead Fortitude is part of the Zombie statblock. It is a special trait the zombie has and part of its statistics. When you use animate dead to animate the corpse, the spell says:

The target becomes a skeleton if you chose bones or a zombie if you chose a corpse (the GM has the creature's game statistics).

So the zombie is using all the statistics from the "zombie" creature, which is the one as linked above, including Undead Fortitude, which means that the animated zombie benefits from Undead Fortitude as usual.

You retain control of the zombie

The conditions for your control over the zombie are said to be, as per the spell description:

The creature is under your control for 24 hours, after which it stops obeying any command you've given it.

Nothing is mentioned about having the zombie use its Undead Fortitude trait. Also note that the zombie, when passing the Undead Fortitude save, is never reduced to 0 HP as the feature says "the zombie drops to 1 hit point instead.". Therefore, as long as the save is passed, you retain control of the zombie as usual for the full 24 hours.

Yes, your zombie gets Undead Fortitude

The answer is 'yes' mainly for the following reason:

There are no secret rules in D&D!

The animate dead spell simply says that you animate a corpse and end up with a zombie.

Since there are no exceptions in the rules that say otherwise, this zombie has the exact statistics given in the Monster Manual (if you look at the DnD Beyond spell entry the link will take you straight to the Zombie stats). This includes the zombie's Undead Fortitude trait.

The zombie will stay under your control for 24 hours (longer if recast) as it says in the spell description. This is because if the zombie is reduced to 0 and makes its Constitution save, it drops to 1 hit point instead.

The 'instead' is important. It means that instead of being destroyed it is at 1 hit point (never actually dropping to 0 hit points at all). Hence, it remains in your control.