HONG KONG—A 53-year-old fitness trainer died on Wednesday after checking into a hospital in Wuhan a little more than a week earlier, said his niece. His family had expected the death certificate to reflect the deadly coronavirus, because as his condition deteriorated, his doctors told his family he was suffering from an untreatable virus in his lungs.

Instead, it recorded “severe pneumonia” as the cause of death, she said. The relatives of two other people who died in separate hospitals in Wuhan this week also described similar...