I'm not fully informed about pumped storage, but twenty minutes is more than enough for most hydroelectric plants to go from 0 to full power, and that's probably the case for pumped storage as well. Eg,
In the case of Cruachan Power Station: “It takes just two minutes for a turbine to run up from rest to generate mode,” says Martin McGhie, Operations and Maintenance Manager at the power station. “It takes slightly longer for the turbines to run down from generate to rest, but whatever function the turbines are performing, they can reach it within a matter of minutes.”
In the case of Cruachan Power Station: “It takes just two minutes for a turbine to run up from rest to generate mode,” says Martin McGhie, Operations and Maintenance Manager at the power station. “It takes slightly longer for the turbines to run down from generate to rest, but whatever function the turbines are performing, they can reach it within a matter of minutes.”
https://www.drax.com/power-generation/in-energy-storage-timi...