Explanations in automated vehicles enhance passengers' understanding of vehicle decision-making, mitigating negative experiences by increasing their sense of control. These explanations help maintain situation awareness, even when passengers are not actively driving, and calibrate trust to match vehicle capabilities, enabling safe engagement in non-driving related tasks. While design studies emphasize timing as a crucial factor affecting trust, machine learning practices for explanation generation primarily focus on content rather than delivery timing. This discrepancy could lead to mistimed explanations, causing misunderstandings or unnecessary interruptions. This gap is partly due to alack of datasets capturing passengers' real-world demands and experiences with in-vehicle explanations. We introduce TimelyTale, an approach that records passengers' demands for explanations in automated vehicles. The dataset includes environmental, driving-related, and passenger-specific sensor data for context-aware explanations. Our machine learning analysis identifies proprioceptive and physiological data as key features for predicting passengers' explanation demands, suggesting their potential for generating timely, context-aware explanations. The TimelyTale dataset is available at https://doi.org/10.7910/DVN/CQ8UB0.