Measurements of durations of nonequilibrium stochastic processes provide valuable information on underlying microscopic kinetics and energetics. Theories for corresponding experiments to date are well-developed for single-particle systems only. Little is known for interacting systems in nonequilibrium environments. We introduce and discuss a basic model for cycle processes interacting with an environment that can be out of thermodynamic equilibrium. We find a surprising richness of cycle time variations with environmental conditions. This manifests itself in unequal cycle times in forward and backward cycle directions, speeding up of backward cycles by interactions, and dynamical phase transitions, where cycle times become multimodal functions of a bias. The model allows us to relate these effects to specific microscopic mechanisms, which can be helpful for interpreting experiments.