Amazon.com Inc.’s drone delivery program has liftoff—from a rural corner of England.

Amazon last week made its first customer delivery by drone, carrying a package containing popcorn and a Fire TV video-streaming device several miles to a two-story farmhouse near Cambridge, U.K., in 13 minutes. A video the company released Wednesday shows a track the drone used to launch, a platform from which employees monitored takeoff, and a landing pad on the customer’s lawn.

The...