This article presents a continuous home telemonitoring system for chronic respiratory patients using 5G connectivity developed in partnership with Vodafone as a part of the 5G Trial in Milan established by the Italian Ministry of Economic Development. The system features a wearable respiratory and activity monitor, an environmental sensor and a pulse oximeter sending the data through a 5G router to a Multi-Edge Computing server, incorporated in the Vodafone 5G infrastructure, where they are stored and accessible for visualization. In particular, activity, respiratory and environmental data are continuously streamed and collected. The solution has been tested on 18 healthy volunteers during non-supervised recordings lasting at least 48 hours. The combination of recognized activities and associated respiratory parameters provided statistically significant variations in breathing patterns between one activity and the other, thus giving more complete information to the clinicians than previously studied telemedicine systems based on spot-checks. In particular, statistically significant differences are found in tidal volume and minute ventilation between horizontal and vertical postures (p < 0.001) and between vertical postures and dynamic activities (p < 0.001); the respiratory rate shows statistically significant differences between horizontal and vertical postures (p < 0.001). Some environmental parameters have different mean values between day and night, such as carbon dioxide (p < 0.001). Trials on patients are needed to further study this telemedicine solution and make it commercially available in the future. The main further technical development suggested is the use of commercial 5G smartphones as routers, in order to make the system usable outside of home settings.