In many real-time applications, timely accessibility to actionable insights is an important requirement. We consider users equipped with sensors and actuators, where the sensors generate updates which need to be processed for driving the actuators. The processing task can be carried out locally or at an edge server located at a common base station (BS). The users access the BS via a fading multiple access channel (MAC) using a non-orthogonal multiple access technique. To measure the timeliness of processed information, we adopt a metric called age of processed information (AoPI), defined as the time elapsed since the generation of the last successfully processed packet available to the user. In this setting, we formulate an average AoPI minimization problem, when users are subjected to average power constraints. The BS has to decide when the users should generate status updates and whether to process them locally or at the edge server. We cast the problem as a constrained Markov decision process (CMDP) and obtain its solution via Lagrangian relaxation. We then obtain a simpler policy via Lyapunov optimization and bound its performance. Via numerical simulations, we finally illustrate properties of the CMDP solution and study behavior of average AoPIs under different policies when problem parameters are varied.