The fifth generation (5G) mobile network delivers high peak data rates with ultra-low latency and massive network capacity. Wireless sensor network (WSN) in Internet of Thing (IoT) architecture is of prominent use in 5G-enabled applications. The electronic healthcare (e-health) system has gained a lot of research attention since it allows e-health users to store and share data in a convenient way. By the support of 5G technology, healthcare data produced by sensor nodes are transited in the e-health system with high efficiency and reliability. It helps in reducing the treatment cost, providing efficient services, better analysis reports, and faster access to treatment. However, security and privacy issues become big concerns when the number of sensors and mobile devices is increasing. Moreover, existing single-server architecture requires to store a massive number of identities and passwords, which causes a significant database cost. In this paper, we propose a three-factor fast authentication scheme with time bound and user anonymity for multi-server e-health systems in 5G-based wireless sensor networks. In our work, the three-factor authentication scheme integrating biometrics, password, and smart card ensures a high-security sensor-enabled environment for communicating parties. User anonymity is preserved during communication process. Besides, time bound authentication can be applied to various healthcare scenarios to enhance security. The proposed protocol includes fast authentication, which can provide a fast communication for participating parties. Our protocol is also designed with multi-server architecture to simplify network load and significantly save database cost. Furthermore, security proof and performance analysis results show that our proposed protocol can resist various attacks and bear a rational communication cost.