Drought is a natural hazard that occurs at many temporal and spatial scales and has severe environmental and socioeconomic impacts across the globe. The impacts of drought change as drought evolves from precipitation deficits to deficits in soil moisture or streamflow. Here, we quantified the time taken for drought to propagate from meteorological drought to soil moisture drought and from meteorological drought to hydrological drought. We did this by cross-correlating the Standardized Precipitation Index (SPI) against standardized indices (SIs) of soil moisture, runoff, and streamflow from an ensemble of global hydrological models (GHMs) forced by a consistent meteorological dataset. Drought propagation is strongly related to climate types, occurring at sub-seasonal timescales in tropical climates and at up to multi-annual timescales in continental and arid climates. Winter droughts are usually related to longer SPI accumulation periods than summer droughts, especially in continental and tropical savanna climates. The difference between the seasons is likely due to winter snow cover in the former and distinct wet and dry seasons in the latter. Model structure appears to play an important role in model variability, as drought propagation to soil moisture drought is slower in land surface models (LSMs) than in global hydrological models, but propagation to hydrological drought is faster in land surface models than in global hydrological models. The propagation time from SPI to hydrological drought in the models was evaluated against observed data at 127 in situ streamflow stations. On average, errors between observed and modeled drought propagation timescales are small and the model ensemble mean is preferred over the use of a single model. Nevertheless, there is ample opportunity for improvement as substantial differences in drought propagation are found at 10 % of the study sites. A better understanding and representation of drought propagation in models may help improve seasonal drought forecasting as well as constrain drought variability under future climate scenarios.