Do Japanese use 順番、順番！ as a command, like in the military?

Unlike 戻る (which is a plain-form imperative), 順番 by itself is not a direct command to make someone move. In this video, the teacher simply reminded the students that there was something called 順番 that they needed to respect. Semantically, this "順番!" is more like "Remember 順番!".

Likewise, we don't say "順番!" in militaristic contexts. A normal drill command to make people line up is "整列!" ("Fall in!").

"順番！"

Taking Turns! = Kids on a slide など (everyone wants to go)

Everyone Goes! = Kids lining up to get vaccine (nobody wants to go)

Used by: Someone in authority (Mom, Teacher, Coach)

順番 is not turn however, it is an idea referring to order, an order of steps, or it could be turns for people. note in kanji we have 書き順.

As for military usage, well perhaps in some instances, though is seem odd that is would need to be shouted at soldiers, as discipline should be their domain.

The sentence in the title 順番、順番！on its own, Turn , turn!, except in an unusual context, such as a response in a classroom as to what this word is, etc, wouldn't likely be used, since it's a noun. For an imperative, a verb meaning to turn and with the word ending for the 'imperative tense' would be used.