Britain’s royal family and television have a complicated relationship. The medium has helped define the modern monarchy: The 1953 coronation of Queen Elizabeth II was Britain’s first mass TV spectacle. Since then, rare interviews have given a glimpse behind palace curtains. The 1981 wedding of 32-year-old Prince Charles and 20-year-old Lady Diana Spencer at St Paul’s Cathedral was a fairy-tale spectacle watched by 750 million people around the world. But the couple separated in 1992, and in 1995 Diana gave a candid interview to the BBC, discussing the pressure of media scrutiny and the breakdown of her marriage. It prompted a wave of sympathy for Diana, seen as a woman failed by an uncaring royal establishment—a pattern that has repeated itself with Meghan Markle. Like Diana before her and Meghan since, Sarah Ferguson was a young woman who had a bruising collision with the royal family. Initially welcomed as a breath of fresh air when she wed the queen’s second son, Prince Andrew, in 1986, she quickly became a tabloid target for her allegedly...