A robust hybrid control strategy for a broad class of hybrid nonlinear processes with actuator constraints and uncertain dynamics is proposed. These variable-structure processes comprise a finite family of constrained uncertain continuous nonlinear dynamical subsystems, together with discrete events that trigger the transition between the continuous subsystems. The proposed control strategy is predicated on the idea of coordinating the hierarchical tasks of lower-level feedback-controller synthesis and upper-level switching logic design. Using multiple Lyapunov functions, a family of bounded robust, nonlinear feedback controllers are initially designed to robustly stabilize the constituent modes of the hybrid process, subject to uncertainty and constraints. The region of guaranteed closed-loop stability is then explicitly characterized for each mode in terms of the magnitude of actuator constraints and the size of the uncertainty. A set of stabilizing switching laws that track the energy evolution of the constituent modes are then derived to orchestrate safe transitions between the stability regions of the constituent modes and their respective controllers, in a way that respects actuator constraints and guarantees robust stability of the overall uncertain hybrid closed-loop system. This hybrid control method is applied through computer simulations to robustly stabilize an exothermic chemical reactor with switched dynamics, model uncertainty, and actuator constraints at an unstable steady-state and to design a fault-tolerant control system for chemical reactors through switching between multiple constrained control configurations.