To call Overlord a Nazi zombie flick is misleading, although that's the go-to phrase for this movie. The Nazi/WWII backdrop is more of a convenience, considering its revisionist alternate history makeup of the army (no one bats an eye at the mixed races and the sergeant is a respected African-American man) and the ease of an indelibly evil adversary. There aren't quite enough zombies to make it a zombie flick either, nor do we have the usual braindead bloodthirsty inanimate bodies scrambling around.
The reason this isn't a zombie movie is because it misses all the beats of its predecessors, even while making references to Moreau and Carpenter. It takes itself completely seriously, almost to a flaw. There are beats in the movie where you expect a joke, but Overlord purposefully elbows those moment aside as if to assure you how seriously we should take it. This is fine, but it then makes it harder to excuse the lapses in judgment the characters make. There are a few too many unzipped bags, syringes used with devil-may-care-consequences, or trusting peeks into darkened cells for any normal person to make. There's an attempt at the theme of not becoming the monsters you're fighting, but it peters out fairly easily. Boyce (Jovan Adepo) is the protagonist here and he serves as a moral compass for the group, supposedly not even able to kill a mouse. But as Overlord progresses, he sometimes unwittingly shows the lack of moral compass for the story, including his readiness to move to violence in one scene. There's no moral quandary or dilemma found there...in fact, you'd find more of that in Shaun of the Dead.
With all that said, however, Overlord is still a fun ride. Part of this is because you come to love the troop, and that's always half the battle. The characters are easily established in the first few minutes, from fast-talking Tibbet (John Maguro) to brooding transfer Ford (Wyatt Russell), and this unfortunately is what so many current horror movies lack and what their weakness is. The livelihoods of these men wouldn't matter a whit otherwise, and thankfully we don't have that problem here. The special effects don't have the same verisimilitude that, say, Saving Private Ryan does, but we don't really need it to. The first sequence of their parachute jump is a completely whipped around, manufactured ride...but it's still exciting and breathless all the same. On par with that is the discovery of the Nazi lab, which is manufactured and breathless in a different way. The eeriness of that scene makes it one of the highlights of the movie, where otherwise things are little too matter-of-fact.
Zombie movies have often been a social commentary on the current events. If Overlord were to serve that purpose, it might be the refreshing recourse of an easy bad vs good and being able to take matters into your own hands on the side of good. The film might not be trying anything at all though, because the strength of Overlord is indeed in the artificiality of it, including its disavowal of the other horrors that can be (and were) created by those with power. Its assurance of no moral grays is vaguely reassuring, and it's a movie that would be better categorized as an action movie that will engage the viewers for its runtime.