In the hour-long programme, which will be aired on Sunday, the Queen speaks candidly about the moment she was crowned, joking that she can not look down while wearing the Imperial State Crown, which weighs 2lbs 13oz (1.28kg), as her "neck would break" if she did so.
The Queen's Coronation at Westminster Abbey in 1953 was one of the great spectacles of the 20th century, a piece of pageantry that transfixed television viewers around the world.
Elizabeth has never given a formal interview during her long reign and Coronation expert Alastair Bruce, whom the queen spoke to for the programme, said their 1-1/2-hour interaction had been a "conversation".
In Graham's autobiography, Just As I Am, the man of faith wrote, "No one in Britain has been more cordial toward us than Her Majesty Queen Elizabeth II". But once you put it on, it stays. "'Above all things, I do think of myself as just a simple Christian'".
The Queen also points out that the Imperial State Crown, worn by her for her speech during the state opening of parliament, and, has been reduced in height since her father King George VI wore it.
The head of state says in an extract from the documentary her crowning was the "beginning of one's life as a sovereign".
She then added, "So, there are some disadvantages to crowns, but otherwise, they're quite important things".
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Clarkson raves about how nice Streep was to meet, even though she's not sure the actress was aware of who she was meeting. You can take my seat any time! "Other than the fact that she's an incredible actor, she's also an fantastic singer".
The stones, including the Black Prince's Ruby from the Imperial State Crown, were placed in the metal box and buried under a sally port, a secured entrance.
The Queen acceded to the throne on 6 February, 1952 when her father died unexpectedly in his sleep at Sandringham in Norfolk.
Other relevations in the programme include how the Queen became stuck with her Coronation dress, which was embroidered in silk with pearls, and gold and silver thread.
"They dug out this fresh, very virgin white chalk and they had to hide it with tarpaulins so when the aircraft flew over at night no clue was given to the German Luftwaffe that anything was going on", Bruce wrote.
Elizabeth II recalls the day she was made Queen of England at the age of 25 in a BBC documentary, describing what it is like to wear the crown.
"Not what they're meant to do", the Queen notes. It was made in partnership with the BBC and Australian Broadcasting Corporation and is part of the Royal Collection Season, a collaboration between the BBC and Royal Collection Trust.
To see the full interview, watch The Coronation on BBC One at 8pm on Sunday.