Recent measurements of the durations of nonequilibrium processes provide valuable information on microscopic mechanisms and energetics. Theory for corresponding experiments to date is well-developed for single-particle systems only. Little is known for interacting systems in nonequilibrium environments. Here we introduce and study a basic model for cycle processes interacting with an environment that can exhibit a net particle flow. We find a surprising richness of cycle time variations with environmental conditions. This manifests itself in unequal cycle times τ+ and τ- in forward and backward cycle directions with both asymmetries τ- < τ+ and τ- > τ+, speeding up of backward cycles by interactions, and dynamical phase transitions, where cycle times become multimodal functions of the bias. The model allows us to relate these effects to specific microscopic mechanisms, which can be helpful for interpreting experiments.