Pryor's mother Margaret was one of the monarch's closest friends, and was a bridesmaid at her 1947 wedding to Prince Philip – who died last month at the age of 99.
During the documentary, Pryor also spoke about the lack of airs and graces among royal friends. She said: "We're all just perfectly ordinary people. The Queen Mother and my granny were very close and the Queen Mother was unbelievably fond of Mummy and her siblings. Mummy called her [the Queen] Lilibet but we call her Ma'am. I mean the Queen would've loved just to have been a country lady with her animals."
Elsewhere in the programme, Lord Ivar Mountbatten echoed Pryor's sentiments.
When asked if they're "a normal family", Lord Ivar said: "Yes, absolutely. That's what everyone seems to forget. We all have our ups and downs, everybody has their own internal arguments - it's the same anywhere."
The documentary also provided a surprising revelation when host Armstrong revealed that he had uncovered a new royal cousin who can now be added to the family tree.