In this article, we evaluate the first experience of computation offloading from drones to real fifth‐generation (5G) operator systems, including commercial and private carrier‐grade 5G networks. A follow‐me drone service was implemented as a representative testbed of remote video analytics. In this application, an image of a person from a drone camera is processed at the edge, and image tracking displacements are translated into positioning commands that are sent back to the drone, so that the drone keeps the camera focused on the person at all times. The application is characterised to identify the processing and communication contributions to service delay. Then, we evaluate the latency of the application in a real non standalone 5G operator network, a standalone carrier‐grade 5G private network, and, to compare these results with previous research, a Wi‐Fi wireless local area network. We considered both multi‐access edge computing (MEC) and cloud offloading scenarios. Onboard computing was also evaluated to assess the trade‐offs with task offloading. The results determine the network configurations that are feasible for the follow‐me application use case depending on the mobility of the end user, and to what extent MEC is advantageous over a state‐of‐the‐art cloud service.