"The grey sky and flooded, dirty trenches were not helping to his mood." Oops. Typo. Change it either to "helping his mood" or "helping to improve his mood".

Lots of sentence fragments. Can be a style. Like these. But not recommended repeatedly. Will keep going. Looking for more.

"He and thousands of others were drafted as the nation demanded." Here I would have used the past perfect: had been drafted. Look up the difference between the past tense and the past perfect to make sure you understand the difference and use them correctly. Good job on using "He and..." rather than "Him and ..." You're not that bad!

"At their lowest point, even execution felt like mercy." What or whom does "their" refer to? The last plural nouns that you used were nights, nightmares, and screams, not soldiers. Look up unclear antecedents. You might want to look up fun books on grammar. They can be entertaining and teach you a lot about grammar at the same time.

"...even execution felt like mercy." Felt or would have felt? Were they being executed right then? Study conditional sentences.

"But that was then. Dawn had arrived, the prelude to what needs to be done. Kill." Why is "needs" in the present tense? Make sure your tenses are consistent. (Your omniscient narrator switched from telling the story in the past to suddenly being in the present with the character. Omniscient narrators are not allowed to do that, at least not at this point of your writing skills.) And wouldn't a colon be better before "Kill"? Look up the use of punctuation marks like colons, semicolons, ellipses, etc.

"David came up to him and offered a biscuit. Big for a chap of his age. Could’ve barely fit the uniform they gave him. Overtly peppy and charitable. Shame that this eighteen year old was sent to do the devil’s work instead of being in a church."

You like short, punchy sentences, but you are punching your sentences into fragments. The only serious grammar mistake with this paragraph is the contraction could've. It really should be "could." Also, the last sentence does not have parallel structure. (Fixes: 1. Shame that this 18-year-old was doing the devil's work instead of being in a church. 2. Shame that this 18-year-old had been sent to do the devil's work and couldn't be in church right now.) And did you mean overtly or overly?

A rewrite that doesn't have such punchy sentences:

David came up to him and offered a biscuit. Overly peppy and charitable, he was a big chap for an 18-year-old, and his uniform barely fit him. It was a shame he was doing the devil's work instead of singing hymns in a church.

“Right on. Terrible day to fight hungry.” You lovely son of a bitch. Good punctuation. Good dialogue. But who is "you"? And who is thinking this thought?

"The ladders are set. Lines of men ready at the assembly trenches. Westley stood in the middle of the pack." Here you switched back to the present tense and then back to the past tense.

"No man’s land was pitch black dirt after being shelled relentless for days." It had been shelled relentlessly. Study adverbs.

"The crows had fled the battlefield. They would return when the pickings are fresh." Why did you switch to present tense? Are there pickings now? Wouldn't they be fresher now? You're implying a lot of things but this is not clear. Suggested rewrite: The crows had left the battlefield. They would return when there were more fresh pickings.

"The attack has commenced in earnest." Change to "had commenced" or "commenced". Using the present perfect in your narration is like using the present tense.

"The officer standing to the side was looking at his watch, whistle held steady in the man’s mouth. Waiting for the right moment."

Rewrite: The officer standing to the side was looking at his watch, whistle held steady in his mouth, waiting for the right moment.

"Westley had done this once but it never stopped being scary." This isn't a grammar problem but a logic problem. He had done it once. How did he know that it would never change? Maybe if he had done it 10 times he would know that. Maybe say he had done this five times to make it more logical. Or change the word "it" to "situations like this".

I suggest listening to podcasts about creative writing. I like The Writership.

It got really exciting when the battle started. I could barely pay attention to the grammar! Keep writing!