Positive-unlabeled learning (PUL) aims at learning a binary classifier from only positive and unlabeled training data. Even though real-world applications often involve imbalanced datasets where the majority of examples belong to one class, most contemporary approaches to PUL do not investigate performance in this setting, thus severely limiting their applicability in practice. In this work, we thus propose to tackle the issues of imbalanced datasets and model calibration in a PUL setting through an uncertainty-aware pseudo-labeling procedure (PUUPL): by boosting the signal from the minority class, pseudo-labeling expands the labeled dataset with new samples from the unlabeled set, while explicit uncertainty quantification prevents the emergence of harmful confirmation bias leading to increased predictive performance. Within a series of experiments, PUUPL yields substantial performance gains in highly imbalanced settings while also showing strong performance in balanced PU scenarios across recent baselines. We furthermore provide ablations and sensitivity analyses to shed light on PUUPL's several ingredients. Finally, a real-world application with an imbalanced dataset confirms the advantage of our approach.