Revealing mistakes: After Grand Fenwick's army sets sail for home, the headlines in several American newspapers talk about war mobilisation. American newspapers would spell it mobilization.
Factual errors: When the Queen Elizabeth encounters the ship carrying the Grand Fenwick troops returning from New York, the ship's officer calls "ship off the port bow." In the next shot, the troop ship is shown off the Queen Elizabeth's starboard bow.
Anachronisms: Grand Fenwick is supposed to have been founded in the 15th century by an English baronet. That title did not exist until 1611, when it was invented by James I-VI.
Errors in geography: The narrator says at the beginning that Grand Fenwick, at 15 square miles, is the smallest country in the world. Monaco and the Vatican are much smaller, so small, in fact, that even put together they are still much smaller than Grand Fenwick. Monaco itself is approximately 1 square mile. Even smaller is the Vatican, who's area is measured in mere acreage.
Continuity: The narrator tells us that the mock version of Grand Fenwick's wine drove the real wine out of the U.S. market in 1959. A news announcer later reports the score of the last game of the 1958 World Series.
Continuity: During the Institute for Physics scene, Professor Kokintz addresses Tully as both "Grand Marshall" and "Mr. Constable", but nowhere in the preceding dialogue is Tully introduced as such. Prof. Kokintz couldn't have known his two titles.