In the opening shot, Hotspur tells a wounded enemy soldier that he is crawling in the wrong direction, using the wind as reference ("That wind you feel is coming from England. Scotland's that way."). However, the position of the setting Sun suggests that Hotspur pointed towards the South as he said "Scotland's that way." And, the smoke in the background belies that the wind is actually coming from behind the crawler.
Hal calls the archers weapon a "long bow", at the time they were only called a war bow.
When Henry V receives the gift from France, (38:29) a candle in the left background falls down at the exact moment the box is set down on the table.
When Prince Henry and Hotspur are first fighting in full armor, modern rubber soles are visible on their feet.
When the first boy outside the camp collecting water gets killed, his modern day boxers are seen under his shirt as he is stabbed.
An important part of the plot is a French man who claims he has been sent to kill the new king. This man is referred to as an assassin. The word assassin didn't enter usage in the English language until at least a century after the historical events depicted in the movie.
The precision of historical facts is not the purpose of this great movie as much as depicting the loneliness and heartlessness in being a powerful king, rising from a very weak situation inherited from his father. For example, to highlight this, in the movie the king's brother Thomas is dead before Henry V becomes king. In reality, Thomas dies in battle only a year before Henry's in 1422. Another example is the character Falstaff, who didn't actually exist in reality, but is a compendium of different historical figures.
Timothée Chalamet's character's accent changes back and forth throughout the movie, sometimes speaking in a British accent, other times in American accent.