The literary movement that this poem's form and elements fit best is Realism. Danny Deever is an 1890 poem by Rudyard Kipling, this is one of the most significant pieces of his early verse. It's composed of four eight-line verses with a dialogue between two voices. Written on the vernacular and with the traditional British poetic format, it belongs to the Victorian literature.
When we talk about realism, we see that the author present familiar things as they are, this means that they depict everyday activities and experiences. The authors try to replicate the true nature of reality and we can see the same in this piece of Kipling as he shows a dialogue between a soldier and his commanding officer about the public hanging of another soldier. It doesn't belong to the Classicism movement because is not about glorifying the Ancient Greek and Roman styles and ideas, it's more to show the life of a soldier and what happens in the military life.
It doesn't belong to the Romanticism movement because is not about an emphasis on emotion and individualism, and also is not about nature ( a current theme in Romanticism). And finally, it doesn't belong to Naturalism movement because even if it's a portrait of reality it doesn't use observation or the scientific method in his poem and in Naturalism the author wants to go further about reality not just to replicate it and this is not the case.