As a rapper, Riz Ahmed has released critically acclaimed albums, and he was featured on the chart-topping “Hamilton Mixtape.” At the same time, he was becoming a leading man in the movies, with roles including a small part in the Star Wars picture “Rogue One” and an extraordinary, Oscar-nominated performance in “Sound of Metal.” Like his previous film, “Mogul Mowgli” is about a musical artist facing a health crisis that could end his career. Ahmed stars as the British-Pakistani rapper Zaheer—stage name Zed—and he co-wrote the film with Bassam Tariq. “It was very much a kind of inward journey,” Ahmed tells David Remnick. “It was very much about holding up a mirror and hoping that in the honesty and the vulnerability of that exercise, people would just connect emotionally—even if they couldn’t necessarily connect to the specificity of the experience. But for those who could connect to that specificity, you would be, like, ‘Jesus Christ, I never thought I’d see that onscreen!’ ”