Just like the title says, how long does it take for this to recharge after you have used it three times?
After you use up all 3 you will have to wait 12 minutes for Urbosa’s Fury to recharge. As with the recharge for Revali’s Gale, you must wait until all 3 uses of the power are gone before the recharge will begin. If you use it twice and have 1 left, the recharge will not begin until you have used the remaining power. Once this is done the recharge will begin and after 12 minutes, all 3 uses will return and you can use this sage power again.
With the introduction of the DLC it is possible to play through the main divine beast and dungeon a second time and it will reduce the cooldown. If you pause the game and view the powerup from the menu, you can see how much time remains before it has recharged.