The performance of deep learning methods critically depends on the quality and quantity of the available training data. This is especially the case for physiological time series, which are both noisy and scarce, which calls for data augmentation to artificially increase the size of datasets. Another issue is that the time-evolving statistical properties of nonstationary signals prevent the use of standard data augmentation techniques. To this end, we introduce a novel method for augmenting nonstationary time series. This is achieved by combining offline changepoint detection with the iterative amplitude-adjusted Fourier transform (iAAFT), which ensures that the time-frequency properties of the original signal are preserved during augmentation. The proposed method is validated through comparisons of the performance of i) a deep learning seizure detection algorithm on both the original and augmented versions of the CHB-MIT and Siena scalp electroencephalography (EEG) databases, and ii) a deep learning atrial fibrillation (AF) detection algorithm on the original and augmented versions of the Computing in Cardiology Challenge 2017 dataset. By virtue of the proposed method, for the CHB-MIT and Siena datasets respectively, accuracy rose by 4.4% and 1.9%, precision by 10% and 5.5%, recall by 3.6% and 0.9%, and F1 by 4.2% and 1.4%. For the AF classification task, accuracy rose by 0.3%, precision by 2.1%, recall by 0.8%, and F1 by 2.1%.