After François has signed the treaty, Henry takes the pen and is seen in profile thinking. Just behind him you can see a blonde extra clearly chewing gum.
Buckingham is shown at his estate called Penshurst. It was originally an estate belonging to the Woodville family which both he and Henry VIII belonged to. Buckingham was the son of Richard and Jaquetta Woodvilles daughter Katherine and Henry was Richard and Jaquetta's descendant through their daughter Elizabeth Woodville and her daughter (Henry's mother) Elizabeth of York.
In the episode Queen Catherine goes to the shrine of Our Lady of Walshingham to prey for a son. Shortly afterward in real life Maria Doyle Kennedy who played Catherine actually did become pregnant with a son.
Ruta Gedmintas's Bessie Blount is terrified that her husband thinks to send her to a nunnery because of her adultery with King Henry; whereas Gedmintas's character in the series Borgia (not to be confused with Tudor's sister series THE BorgiaS), actually fights to go to and remain in a nunnery because of her lover (Cesare Borgia) killing her husband over their affair.
King Francis tells Henry VIII that he looks French with the beard. Besides some more antiquated French ancestry Henry's great grandmother Jaquetta Woodville was French.