NANJING, China—At a cancer conference in Chicago in June last year, a little-known Chinese startup stunned researchers with early results showing its experimental gene therapy was abating an aggressive form of blood cancer in patients back home.

Six months later, the startup, Nanjing Legend Biotechnology Co., received $350 million from Johnson & Johnson for the global rights to co-develop and market the experimental treatment. In May, the U.S. Food and Drug Administration approved its testing on Americans, making it the first...