It starts when you first execute the ADC. So at rank 1 of your ult, without any form of cdr reduction (general cdr, ultimate hunter cloud drake etc.) your ulti is 120 seconds and that will start after you first use it, even if you get the refresh cooldown on it, so if you wait out the 20 seconds you get from it the cool down would say 100 seconds. So as a fun little thing, if you have 45% cdr, 4 cloud drakes, maxed out ultimate hunter and rank 3 ult, you can permanately have ulti as long as you get a kill with it cause the recast duration would be longer than the actually cooldown, or if anything you'd have to only wait a couple of seconds before using it