In reality, Fagan and the Queen didn't speak for very long.
According to “The Crown”, during his second break-in, Fagan tiptoed into Queen Elizabeth’s bedroom, threw open the curtains and woke her up. However, the real Fagan told The Sun that he didn't wake up the Queen. "She was wide awake when I got in there," Fagan said.
The Washington Post reports that, "On screen, [Fagan] wants to chat with the Queen about the tough economic policies of Prime Minister Margaret Thatcher. In real life, it was probably the drugs—specifically magic mushrooms—that made him do it."
“I forgot you’re only supposed to take a little handful,” he told the Independent in 2012. “Two years later, I was still coming down. I was high on mushrooms for a long, long time.”
While the show depicts Fagan having a lengthy conversation with the Queen on the night of the break in, in reality the exchange was brief. "I pulled back the curtain and she said, ‘What are you doing here?’" Fagan has said. After sitting down on the bed, Fagan reported that she added "I’ll be back in a minute" and quickly left the room. Fagan was then taken down the hall and given a whiskey by a footman, and was arrested shortly thereafter.
Furthermore, the episode shows that Fagan wanted to break into the palace so he could share his concerns with the Queen about the then-Prime Minister, Margaret Thatcher. However, that is not true. According to The Sun, "Fagan was feeling depressed after his wife Christine ran off and left him to bring up their four young children, so he set off to see if Her Majesty could solve his problems."