Recent probabilistic model checking techniques can verify reliability and performance properties of software systems affected by parametric uncertainty. This involves modelling the system behaviour using \emph{interval Markov chains}, i.e., Markov models with transition probabilities or rates specified as intervals. These intervals can be updated continually using Bayesian estimators with imprecise priors, enabling the verification of the system properties of interest at runtime. However, Bayesian estimators are slow to react to sudden changes in the actual value of the estimated parameters, yielding inaccurate intervals and leading to poor verification results after such changes. To address this limitation, we introduce an efficient interval change-point detection method, and we integrate it with a state-of-the-art Bayesian estimator with imprecise priors. Our experimental results show that the resulting end-to-end Bayesian approach to change-point detection and estimation of interval Markov chain parameters handles effectively a wide range of sudden changes in parameter values, and supports runtime probabilistic model checking under parametric uncertainty.