The COVID-19 pandemic has changed the world. Today, the use of information and communications technology (ICT) in support of education, medicine, business, and administration has become a reality practically everywhere. In particular, the eHealth (digital Health) sector is on the cusp of a revolution, fueled by the worldwide health emergency due to the spread of the new coronavirus. With a view to developing new sixth-generation (6G)-oriented architectures, advanced eHealth services like telemonitoring would benefit from the support of technologies that guarantee secure data access, ultralow latency and very-high reliability targets, which are hardly achievable by the fifth generation (5G). This is the reason why this work proposes an eHealth system architecture, in which low-latency enabling technologies like Device-to-Device (D2D) communications and multiaccess edge computing (MEC) are integrated and supported by security mechanisms for an optimal management of sensitive health data collected by Internet of Medical Things (IoMT) devices. A preliminary evaluation of the proposed framework is provided that shows promising results in terms of data security and latency reduction.