It happens after Initiative is rolled ...
Battle Cry, General's Cry, and similar abilities allow you to demoralize someone. The important link between demoralize and initiative is that demoralize inflicts the Frightened condition, which in part says:
"You take a status penalty equal to [the Frightened] value to all your checks and DCs. "
Does Frightened influence an initiative role? If you used a check as part of your initiative, then yes.
As covered in this answer to a related question, initiative is a step in the encounter process. Initiative is not a check, although we often use checks to determine initiative. If you use check (such as Perception check) to determine your initiative, then you should include the Frightened penalty.
... but ask your GM.
But what about timing? The text of Battle Cry says it happens when you roll initiative. Does the timing of Battle Cry allow it to affect co-occurring initiative rolls?
Maybe. Ask your GM.
The structure an encounter says that the first step of an encounter is rolling initiative. There is no process which controls the order that initiative is rolled.
One interpretation is that all initiatives happen at the same time. So an effect like Battle Cry would be expected to decrease initiative. This is how initiative is handled at the tables I have been at, though I have never seen someone use Frightened in this way. A second interpretation could be that once someone declares their initiative it is "locked in" and can't be changed. To be sure, ask your GM.