Renewal processes are generalizations of the Poisson process on the real line whose intervals are drawn i.i.d. from some distribution. Modulated renewal processes allow these interevent distributions to vary with time, allowing the introduction of nonstationarity. In this work, we take a nonparametric Bayesian approach, modelling this nonstationarity with a Gaussian process. Our approach is based on the idea of uniformization, which allows us to draw exact samples from an otherwise intractabl...