Taxis provide an important market for electric vehicles (EVs), but long charging durations and limited charger availability have prevented rapid adoption. Leveraging over two weeks of high-resolution GPS and battery data from almost 20,000 EVs in the all-electric Shenzhen taxi fleet, we analyze the potential to improve fleet-wide operations by optimizing both the location and timing of vehicle charging. We construct machine learning models to predict travel time, queuing time at charging stations, and charge consumption by time of day. Contrary to the emphasis on charging station siting in the literature, we find that optimizing charging locations would have a relatively limited impact. Instead, providing drivers with better real-time information about queuing times at charging stations, and enabling flexibility in battery charge during shift changes could reduce down-time per vehicle by over 30 minutes per day, while increasing the number of economically viable charging stations by over 50%. Moreover, taking full advantage of break periods and nighttime to charge could reduce downtime per vehicle by over one hour per day, reducing revenue losses due to charging by roughly 90%. These results are verified with evidence from real-time charging station data and driver shift-change data. Policy recommendations from this study include establishing citywide open data platforms to integrate real-time data on vehicle trajectory, battery charge, and charger availability, and providing drivers and companies with training on best charging practices. As a number of cities worldwide move toward fully electrified taxi fleets, this analysis has large-scale implications for decarbonized, cleaner urban areas.