The political thriller “Bodyguard” proved so popular in the U.K. that real-life government officials weighed in on its plots and characters. Now Netflix is betting that the rest of the world wants to watch.

The streaming-video giant, which said last week that it ended the third quarter with 137 million global subscribers, is releasing all six episodes of the television show in 190 countries on Wednesday. It stars Richard Madden (perhaps best known to American viewers as Robb Stark from “Game of Thrones”) as David Budd, a police...