Revealing mistake: After Blackadder accidentally kills Smedly with one of the suicide pills, he starts ranting about how useless Smedly is as a hero, because he was dumb enough to fall for the old 'poisoned cup routine.' As he does so, you can see the supposedly dead Smedly behind him, still breathing normally.
Deliberate mistake: There are several historical dates wrong in this episode. For a start, Prince George didn't become Prince Regent until 1811, and yet this is set before the Battle of Trafalgar in 1805 (as Black Adder gives Wellington some advice on it). Similarly, Wellington didn't get the title "Duke of Wellington" until 1814. Again inconsistent with being pre-Trafalgar.
Factual error: When Blackadder has disguised himself as the Prince and is speaking to Wellington, Wellington says, "I have with me here the most recent briefs from my General in the field". Blackadder, deliberately misunderstanding, replies "If you would like to pop them in the laundry basket". Blackadder is referring to the short, tight style of men's modern underwear, known as briefs, which were not invented until the 1930s.