This paper presents the design and development of a flexible hyper-distributed IoT–Edge–Cloud computing platform for real-time Digital Twins in real logistics and industrial environments, intended as a novel living lab and testbed for future 6G applications. It expands the limited capabilities of IoT devices with extended Cloud and Edge computing functionalities, creating an IoT–Edge–Cloud continuum platform composed of multiple stakeholder solutions, in which vertical application developers can take full advantage of the computing resources of the infrastructure. The platform is built together with a private 5G network to connect machines and sensors on a large scale. Artificial intelligence and machine learning are used to allocate computing resources for real-time services by an end-to-end intelligent orchestrator, and real-time distributed analytic tools leverage Edge computing platforms to support different types of Digital Twin applications for logistics and industry, such as immersive remote driving, with specific characteristics and features. Performance evaluations demonstrated the platform’s capability to support the high-throughput communications required for Digital Twins, achieving user-experienced rates close to the maximum theoretical values, up to 552 Mb/s for the downlink and 87.3 Mb/s for the uplink in the n78 frequency band. Moreover, the platform’s support for Digital Twins was validated via QoE assessments conducted on an immersive remote driving prototype, which demonstrated high levels of user satisfaction in key dimensions such as presence, engagement, control, sensory integration, and cognitive load.