Security is critical in the deployment and maintenance of novel IoT and 5G networks. The process of bootstrapping is required to establish a secure data exchange between IoT devices and data-driven platforms. It entails, among other steps, authentication, authorization, and credential management. Nevertheless, there are few efforts dedicated to providing service access authentication in the area of constrained IoT devices connected to recent wireless networks such as narrowband IoT (NB-IoT) and 5G. Therefore, this paper presents the adaptation of bootstrapping protocols to be compliant with the 3GPP specifications in order to enable the 5G feature of secondary authentication for constrained IoT devices. To allow the secondary authentication and key establishment in NB-IoT and 4G/5G environments, we have adapted two Extensible Authentication Protocol (EAP) lower layers, i.e., PANATIKI and LO-CoAP-EAP. In fact, this approach presents the evaluation of both aforementioned EAP lower layers, showing the contrast between a current EAP lower layer standard, i.e., PANA, and one specifically designed with the constraints of IoT, thus providing high flexibility and scalability in the bootstrapping process in 5G networks. The proposed solution is evaluated to prove its efficiency and feasibility, being one of the first efforts to support secure service authentication and key establishment for constrained IoT devices in 5G environments.