It occurs at a specific point in the storyline; Once you start quest "State of War", after completing the first step of that, you should also get quest "Under One Banner" - once that is active, eventually someone from HQ will buzz you to tell you there's someone waiting at HQ to talk to you, that is when you should start meeting people of the factions you're eligible to "ally" with.
This presently only affects the story for some choices later on, it doesn't let you recruit their troops into your actual army yet.
And yeah I don't think the Embassy room in the HQ actually does anything mechanically. I've never even seen a single NPC use it, either, but I practically never see anyone working at my HQ go down to that entire level anyway, even with that bar/dining hall the size of Hangmans Alley or all the Living Quarters I threw in just because some rooms had no other choice.