Near a bifurcation point, the response time of a system is expected to diverge due to the phenomenon of critical slowing down. We investigate critical slowing down in well-mixed stochastic models of biochemical feedback by exploiting a mapping to the mean-field Ising universality class. We analyze the responses to a sudden quench and to continuous driving in the model parameters. In the latter case, we demonstrate that our class of models exhibits the Kibble-Zurek collapse, which predicts the scaling of hysteresis in cellular responses to gradual perturbations. We discuss the implications of our results in terms of the tradeoff between a precise and a fast response. Finally, we use our mapping to quantify critical slowing down in T cells, where the addition of a drug is equivalent to a sudden quench in parameter space.