What day is it today? – He asked Ann in a hope that it was just a dream.- Today is the 22nd of June, 1941. Sunday.
My suggestion for this dialogue (keeping in mind that from Ann's point of view, everything seems "normal"):

"What day is it today?" he asked Ann, hoping that it was just a dream.
"It's Sunday," she answered.
"No, um... I meant to ask, what's the DATE today?"
"Oh, it's the 22nd of June."
"But what YEAR is it?" he persisted.
She gave him a strange look. "1941, of course."

I think our police will contact the USA embassy in Minsk, I think Mr. Roosevelt will help you out. – Ann smiled.

- Okay, – Peter replied and automatically corrected her, – Mr. Obama. Barack Obama is the current President of the United States.
Note that the abbreviation "USA" is rarely or never used as an adjective; we'd say "the US embassy consulate" or "the American embassy consulate" instead. However, I think it's a good idea to introduce some small errors in Ann's English (for example, occasionally forgetting to use "a" or "the") so she doesn't sound exactly the same as Peter, who's a native speaker. Also, there are "Roosevelts" in America with no relation to the two presidents of that name, so Peter wouldn't immediately think that "Mister Roosevelt" refers to FDR:

"I think our police will contact USA consulate in Minsk, and Mr. Roosevelt will help you out," Ann smiled.

"Okay, I'd like to contact the American consulate," said Peter, gently correcting her. "And Mr. Roosevelt, I assume, is the US consul there?"

"What? Oh, no," she laughed, "I have no idea at all what the man who works at USA... at the American consulate is called. I was talking about your President in Washington, Franklin Roosevelt!"

Peter hesitated -- was she making some kind of joke? Aha, he thought he'd guessed it: she was talking about Franklin Roosevelt's face on the ten-cent coin. To show that he understood the joke, he chuckled, "Ha-ha... I'm not sure that a Roosevelt dime would cover the cost of a trans-Atlantic telephone call to Barack Obama!"

She gave him a completely blank look. "I'm sorry..." she began, "I don't know what is -- a Roosevelt 'da-yeem'? -- but 'Barockobama' is the name of your home city, I guess?"

Now he gave her an equally blank look.

Still slightly embarrassed at not understanding the word 'dime', she offered quickly: "I suppose that 'Barockobama' comes from an old Indian name, like 'Massachusetts' and 'Mississippi'?"

"Mr... Barack... Obama," he said in a slow daze, "is the current President of the United States."

"I don’t think so," she said smiling, "as far as I know it’s Franklin D. Roosevelt..."