Let me see if I can describe what I have encountered as clearly as possible.
What I have done is run the recruitment missions that are available and they go into what I assume is a cooldown (the buttons remain greyed out)
Let me preface this with the fact I have NOT verified that a specific elapsed period of time has occurred before clicking again.
I come back to the recruitment officer and all the buttons are greyed out still. Every once in a while I can click on at least the general recruitment button and it will allow me to take the mission, despite the button being grey.
So far I have only encounted this with the general recruitment (that is the top most button correct?)
Again I do not know if the cooldown period has occurred and the button is just not displaying as clickable properly or if I am jumping some queue and getting an early crack at the mission again.
The recruitment missions on the Academys have a CD somehting like 2d from start of the Culture Exchange/General Recruitment mission and i think 4d from start for the Tactical/Science/Engin/Civil Missions.
The mission times and cooldowns are funky. The general recruitment mission takes 1 day the others take 2 days. I'm not sure what the cooldowns are.
Sometimes it seems like the other missions don't come off cooldown until I restart the general recruitment mission. But, today, all 5 of them were available and I started them all.
I would love to hear- definitively- what the cooldown times are.