Cooldown runs the moment you cast it. This is true for all prestige across all of coop.
So the only time when the player is “wasting” that time is when they “save it” for the next “problem”. While this isn’t necessarily wrong, it is the very reason why the general recommendation is to use them off cooldown whenever you can.
The above rule isn’t an one-rule-for-all-situations but certainly covers more situations than “saving it”. So for Alarak’s Death Fleet, after that 10min coolup, if the player doesn’t use it at 10min but instead used it at 13min, then the cooldown only start to apply at that 13th minute for the next 6min. And that’s what lends this player to having their 2nd Death Fleet at 19min (13 + 6) at its earliest, whereas the 10min cast player would have one at 16min. Overall 3min wasted so to speak. Of course, the idea is this player needs to situate him/herself to a position where using Death Fleet at 10min is useful.
The implication of these 2 different timings are often what makes or breaks a game (as these situations only occur usually with less experienced players). So the idea of “saving calldown” is usually not very applicable, except in mutational situations or specific speedrun tactics.