Mainly I focused on corrections to Peter's dialogue (both spoken and internal), since it's important that (a) he should SOUND like an American from 2014, in his grammar and word-usage; and (b) he should REACT and THINK like an American from 2014.
For example, an American from 2014 would find Ann's use of the "obsolete" phrase Soviet Union to be much more odd than the word Belarus -- so I changed the dialogue a little to reflect this.

I also had a suggestion about the miles/kilometers problem that Ramil brought up. Americans who jog/run as an amateur sport are really quite accustomed to using "kilometers" for long-distance races -- even though they invariably use "miles" when driving in cars! And we say things like "How many grams of fat in a half-pound of butter?". And we buy milk by the half-gallon, but Cola-Cola comes in 2-liter bottles, etc. In short, Americans do find it natural to use metric units SOMETIMES, but only in certain specific contexts.

‘You are in the Soviet Union -- Belarus,' she said with a slight accent.

‘Oh god, Belarus. I can’t believe it.’ 'The Soviet Union? I can't believe it,' he said aloud, while thinking to himself: 'Did she really say Soviet Union?! Oh, great, just my luck -- I meet a beautiful foreign woman who speaks English, but she's some kind of political fanatic who's still fighting the Cold War in her head!'

‘What’s wrong?’ she asked.

‘Nothing, sorry,’ he said, thinking: ‘Everything’s wrong! Yesterday I was at home on the other side of the globe and now I’m here in the middle of nowhere and I don’t know how I got here and how to get out’.

Almost as if reading his mind, Ann laughed: 'Don't worry, it's not as if you're lost in the middle of Amazon jungle! True, it's a quite small village, but we're only about 110 kilometers due North from Minsk.'

'I'm not exactly sure how far that is in miles,' she added apologetically, after a short pause.

Like most Americans, Peter used the metric system only when absolutely necessary. But since he had run quite a few "10-K mini-marathons" in his younger years -- the doctors had recommended jogging as physical therapy after the accident -- he had no trouble doing this conversion in his head. '110 kilometers would be something like 70 miles, give or take,' he explained. 'So that should be only about an hour's drive by car.'
Of course, an American from 2014 would have no idea of the road conditions in rural Belarus back in 1941 -- so traveling 110 km in one hour would seem reasonable to Peter!

The old man told said something to Ann...
Oddly enough, it would also be fine to say "he told Ann something" or "he told the story to Ann," but "he told something to Ann" sounds very awkward. And "he said Ann something" or "he said Ann the story" or "he said the story to Ann" are all impossible, but "he told Ann the story" is fine. So the word-order is heavily dependent on the choice of words.

‘United States, ma’am,’ he replied, thinking to himself: ‘It looks like the interrogation began is starting sooner than I expected. It had to happen sooner or later. it must have happened.’
‘Can you take bring them with you when you come to see me tomorrow, would you?’
'I see,’ Peter drawled said slowly as a thought stroke across struck his mind
To me, saying "drawled" in this particular context would make it sound like Peter was sarcastically pretending to be simple-minded (as though he's an atheist rudely mocking her belief in "divine providence").

Nikolai told Peter to follow him, at least that’s how Peter interpreted the abracadabra he said [b]gobbledygook he spoke[/s]
Note that in English, turkeys say "gobble-gobble" -- and domestic turkeys (but not the wild ones) are stereotypically thought to be the stupidest of all birds. So this is probably where the Americanism "gobbledygook" comes from.

Nikolai took out of the old frayed scarred cupboard
The word "frayed" is really only used with soft, fibrous materials such as rope, thread, fabric, clothing, etc. -- not with items made of wood. For things made of metal, "dented" is often the best choice. "Scarred" or "battered" are appropriate for an old wooden cupboard, although you could say "splintery" for wooden furniture that has been FRESHLY damaged.

Considering the bombers flying overhead, the border has got to be is in a stone’s throw away ... When will you are you going to evacuate?
There's no preposition "in" in the expression "to be a stone's throw away"; "has got to be" sounds more natural here than "is"; and expressing the simple-future tense with "going to" instead of "will" helps to emphasize the urgency here. (If you like, you can also replace "got to" with "gotta" and "going to" with "gonna" -- although since Peter knows that English isn't Ann's first language, he might naturally speak to her a little more carefully.)

This is what the regular army should do, not you.’
You might also consider changing "army" to "army or navy" -- it would subtly hint that Peter doesn't know Belarus's geography, and therefore has no idea how far they are from the seacoast, where the navy (and/or marine infantry) would operate.

[QUOTE]‘What if they WILL DO treat the civilians bad?’ Peter asked looking in Ann’s eyes.

By strict rules of grammar, one should say "badly" here, not "bad." But under the circumstances, "bad" sounds perfectly normal from a native speaker (although informal). However, "will" is simply not possible here, and one must say "do". (However, in Ann's sentence before this, both "will not" and "do not" are correct, and in fact there's very little difference in meaning: "I hope the Germans don't/won't treat the civilians real bad.")

‘Foreigners; who the hell then I am I, then?’ he thought,
You could also put "then" at the beginning of the clause: "then who the hell am I?", with no need for a comma.

‘Why [b]did[b] God or whoever played this mean joke slipped by slipping me into this place?’,
The above changes in the verb forms are essential. But I might also suggest "Why did God, or some kind of genie, or freaking space-aliens, or whoever it was, play..." (to emphasize Peter's theological uncertainty at this point -- he only knows that something paranormal is involved!). And although the verb "to slip" is correct, it sounds slightly textbook-ish here -- I would recommend "to stick" as more natural.

‘There are is not a single clock at in this house
...I saw camerade comrade Somoff...
As is often the case, the English spelling is inconsistent: "They felt a sense of camaraderie with their comrades" -- although both words come from the same French origin!

Somoff? Who is it that?
"Who is it?" would be the normal question if Somoff (or someone else) were knocking at the door.

There were a dozen of soldiers in the truck's bed with strange tommy-guns (semi-)automatic rifles in their hands
"A dozen soldiers" but "Dozens of soldiers" -- take your pick.

The term "tommy-guns" sounds rather old-fashioned -- an actual American soldier from 1941 might have used it, but not a time-traveler from 2014!

Because it was mentioned early in the story that Peter has some experience as a hunter, I assume that he would probably know the difference between a "pistol" and a "rifle" and a "machine gun", or between a "semi-automatic" and "automatic," etc. If you want to emphasize that Peter is ignorant of firearms, however, I would change "tommy guns" to the incorrect "machine guns" -- which is quite a common mistake. Or you can simply say "strange guns," without being more specific.

...in a uniform similar to the sergeant’s... Sergeant Somoff answered and all of a sudden...
Throughout this whole paragraph, there should be a "the" before "sergeant," unless "Sergeant" (capitalized) is used as a title before a surname such as "Somoff."

only by chance not torn tearing it off the old rusty hinges
seemingly still absent-minded in a daze
"Absent-minded" suggests the meaning "slightly forgetful" or "not quite paying close attention," etc. -- not that someone is nearly catatonic from shock.

whispered ‘It’s empty, move get into it’.
just grabbed her, lifted her (up) and slid her into the barrel
Repeating the pronoun "her" sounds better here. And to me, "lift up" provides a better rhythm here than "lift" alone, but the "up" isn't required by grammar.

‘Wasn’t that barrel empty?’ he was a little scared ... ‘What was in there?’
‘Mushrooms’, she said, fixing up her hair. ‘Pickled.’
This line from Ann was delightfully written! I just thought that it sounded even better with a full-stop and a capital "P", therefore breaking it into two separate (and amusingly laconic!) sentences.