In this episode the Queen appears in court against Homer Simpson. In real life the reigning British Monarch is protected from prosecution and also cannot be called to give evidence in a case of criminal law.
Queen Elizabeth's luggage bears the initials "HRH", for "Her Royal Highness" (which Homer thinks stands for "Henrietta R. Hippo"). This is a mistake; the Queen's husband and children were styled "His/Her Royal Highness", but the Queen herself was "Her Majesty".
Treason is a crime against one's own country, so as an American, Homer couldn't be charged with treason against Britain. Also, the UK formally abolished the death penalty for treason in 1998.
The tunnel Homer uses to escape from the Tower of London, supposedly originally used by Sir Walter Raleigh, comes up in the Queen's bedroom in Buckingham Palace. Buckingham House (as it originally was) was not begun until 1703, nearly a century after Raleigh's death, and most of the present building dates to the 19th century. The two sites are also roughly three miles apart.
Once Burns claimed his $1000 bill, Bart could've taken some of his museum profits to buy a new one. Alternately, the Simpson family could've used museum profits to reimburse Burns for the original bill.
Marge claims that in England miles are called kilometres. Although metric measures are indeed used in the UK, distances (on road signs, for example), are generally given in miles.
Lisa mispronounces JK Rowling's name, with the first syllable rhyming with 'cow' instead of 'low'.