LONDON—On the day his police guards collected him for a televised “confession,” Peter Humphrey says, he had swallowed a tranquilizer pill after a panic attack during the night.

The British corporate investigator had been held along with Yu Yingzeng, his American wife and business partner, in 2013 in a case involving one of their clients, the drugmaker GlaxoSmithKline PLC. Police at the Shanghai No. 1 Detention Center had warned him he would be meeting with journalists. “If you handle this well you’ll get lenient treatment,”...