Autonomous systems operate in environments that may change over time. An example is the control of a self-driving vehicle among pedestrians and human-controlled vehicles whose behavior may change based on factors such as traffic density, road visibility, and social norms. Therefore, the environment encountered during deployment rarely mirrors the environment and data encountered during training -- a phenomenon known as distribution shift -- which can undermine the safety of autonomous systems. Conformal prediction (CP) has recently been used along with data from the training environment to provide prediction regions that capture the behavior of the environment with a desired probability. When embedded within a model predictive controller (MPC), one can provide probabilistic safety guarantees, but only when the deployment and training environments coincide. Once a distribution shift occurs, these guarantees collapse. We propose a planning framework that is robust under distribution shifts by: (i) assuming that the underlying data distribution of the environment is parameterized by a nuisance parameter, i.e., an observable, interpretable quantity such as traffic density, (ii) training a conditional diffusion model that captures distribution shifts as a function of the nuisance parameter, (iii) observing the nuisance parameter online and generating cheap, synthetic data from the diffusion model for the observed nuisance parameter, and (iv) designing an MPC that embeds CP regions constructed from such synthetic data. Importantly, we account for discrepancies between the underlying data distribution and the diffusion model by using robust CP. Thus, the plans computed using robust CP enjoy probabilistic safety guarantees, in contrast with plans obtained from a single, static set of training data. We empirically demonstrate safety under diverse distribution shifts in the ORCA simulator.