What are the cooldowns for the Divine Beasts' Abilities?

The numbers are indeed stacks. The ability will not start restoring until all stacks are used.

• Zora ability is 1x (with a cooldown of 24 minutes)
• Gerudo ability is 3x (with a cooldown of 12 minutes)
• Rito ability is 3x (with a cooldown of 6 minutes)
• Goron ability is 3x (with a cooldown of 18 minutes)

There seems to be a little confusion on this in the other answer, even though it was accepted already. Here it shows the names of the abilities.

• Rivali's Gale - 6 minutes
• Urbosa's Fury - 12 minutes
• Daruk's Protection - 18 minutes
• Mipha's Grace - 24 minutes

Note that all except Mipha's Grace have 3 uses before the cooldown comes into effect.

These times correspond to the day-night cycle of the game. The game's clock runs one minute for each second in real-world time, so the game has a 24 minute day-night cycle. Each of the powers correspond to 1/4, 1/2, 3/4, and 1 full day to recharge. Fast forwarding by sitting by a fire or sleeping in a bed does not speed this along.

Update for how the Champion's Ballad DLC changed things

The DLC will allow you to unlock story missions related to each of the divine beasts. The completion of each Champion's quest will cause their power to decrease to 1/3 the original time, so the new cooldown times are: Rivali's Gale - 2 minutes, Urbosa's Fury - 4 minutes, Daruk's Protection - 6 minutes, Mipha's Grace - 8 minutes