Your allies are instantly available for deployment the moment they log in and will be displayed with '1 hour remaining' the whole time they play Fantasica (either by doing a quest or through training or buying boxes).
Once they stop doing anything in Fantasica or log out, the timer will go down from 1 hour to 0. Once the timer reaches zero, you cannot deploy those inactive allies and you will have to wait until they return to the game.
Your player level also affects how long your allies will be available for deployment. If your player level is lower than your ally's player level, then you will have less time to use them should they log out. For example, a Lv35 player will only have 13 minutes to use a Lv100 ally the moment they stop playing.
There are more answers on Ally Use here: