Since DN, module cool-downs in WS are persistent (even if you jump your ship out of the gate, wait for the modules to refresh in YS, and jump back into the WS the modules will still be on cool-down).
The issue is, that without jumping back into the WS with the module in question equipped, it’s not possible to see how much time is left on the cooldown.
What’s needed, is some way to see, without jumping back in with multiple different modules equipped, or repeatedly replaying TM history to see what it was when you last had it equipped. Whether that is best done on a page listing all the current team statuses, or through a personal ship-linked status display will require some thought/discussion.
For my part, I think since it’s a team match, it would be beneficial (especially to WS commanders) if there was a new page introduced, with a list of the players in your WS team, their 3 ship statuses and all available modules statuses.
(From an espionage perspective, access to this could be limited to officers, like the FS is, and all member/senior member players are limited to their own ship & mod statuses, or not, if it’s not considered to be any more of a risk than the current setup.)