Traditionally, process control systems utilize dedicated, point-to-point wired communication links using a small number of sensors and actuators to regulate appropriate process variables at desired values. While this paradigm to process control has been successful, chemical plant operation could substantially benefit from an efficient integration of the existing, point-to-point control networks (wired connections from each actuator/sensor to the control system using dedicated local area networks) with additional networked (wired or wireless) actuator/sensor devices. However, augmenting existing control networks with real-time wired/wireless sensor and actuator networks challenges many of the assumptions made in the development of traditional process control methods dealing with dynamical systems linked through ideal channels with flawless, continuous communication. In the context of control systems which utilize networked sensors and actuators, key issues that need to be carefully handled at the control system design level include data losses due to field interference and time delays due to network traffic. Motivated by the above technological advances and the lack of methods to design control systems that utilize hybrid communication networks, in the present work, we present a novel two-tier control architecture for networked process control problems that involve nonlinear processes and heterogeneous measurements consisting of continuous measurements and asynchronous, delayed measurements. This class of control problems arises naturally when nonlinear processes are controlled via control systems based on hybrid communication networks (i.e., point-to-point wired links integrated with networked wired/wireless communication) or utilizing multiple heterogeneous measurements (e.g., temperature measurements which can be taken to be continuous and species concentration measurements which are fed to the control system at asynchronous time instants and frequently involve delays). While point-to-point wired links are very reliable, the presence of a shared communication network in the closed-loop system introduces additional delays and data losses and these issues should be handled at the controller design level. In the two-tier control architecture presented in this work, a lower-tier control system, which relies on point-to-point communication and continuous measurements, is first designed to stabilize the closed-loop system, and an upper-tier networked control system is subsequently designed, using Lyapunov-based model predictive control theory, to profit from both the continuous and the asynchronous, delayed measurements as well as from additional networked control actuators to improve the closed-loop system performance. The proposed two-tier control architecture preserves the stability properties of the lower-tier controller while improving the closed-loop performance. The applicability and effectiveness of the proposed control method is demonstrated using two chemical process examples.