Whether or not the holiday is celebrated in English-speaking countries, I disagree with the English translations that the book gives for the Russian phrase С Первым мая!, namely:
"Happy May Day!" would be much, much more natural. "May Day greetings to you" sounds too formal, and English speakers generally don't say "Congratulations" on the occasions of annual holidays like Christmas, New Year, Easter, Mother's Day, 4th of July in America, etc. Instead, we say "Happy [name of holiday]!", unless it's Christmas and you're American, in which case "Happy" is changed to "Merry," for some reason.Congratulations! (May Day greetings to you!)
The phrase "Congratulations" is used when someone gets married, has a baby, graduates from school, is promoted to a higher position at work, buys a new home, etc. -- but not for holidays.
We also sometimes say "Congratulations" for a person's birthday -- though more often for young children than for adults. So "Congratulations, you're six years old now!" is quite commonly heard, but "Congratulations, you're 37!" would be rather unusual!