We study the dynamics of the out-of-equilibrium nonlinear q-voter model with two types of susceptible voters and zealots, introduced in Mellor et al. [Europhys. Lett. 113, 48001 (2016)EULEEJ0295-507510.1209/0295-5075/113/48001]. In this model, each individual supports one of two parties and is either a susceptible voter of type q_{1} or q_{2}, or is an inflexible zealot. At each time step, a q_{i}-susceptible voter (i=1,2) consults a group of q_{i} neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. This model violates detailed balance whenever q_{1}≠q_{2} and is characterized by two distinct regimes of low and high density of zealotry. Here, by combining analytical and numerical methods, we investigate the nonequilibrium stationary state of the system in terms of its probability distribution, nonvanishing currents, and unequal-time two-point correlation functions. We also study the switching time properties of the model by exploiting an approximate mapping onto the model of Mobilia [Phys. Rev. E 92, 012803 (2015)PLEEE81539-375510.1103/PhysRevE.92.012803] that satisfies the detailed balance, and we outline some properties of the model near criticality.