3.5 fear / turning effects timing

So, we had a discussion about this today in a DnD session. We had some undead that were turned and running, and the DM had them move to run away instantly, rather than wait until their initiative order.

My thought is that the enemies need to wait until their next turn in the initiative order to flee, while my DM says they would move immediately on the cleric's turn.

This has branched out to a discussion of fear and panicking in general, and we are unable to find anything in the rules or online that explicitly states one way or the other, and so I'm asking here.

Just wanting to see if there's something we missed and / or there's a consensus one way or the other. We both DM, and often have rules discussions and disagreements, but this is the first time we haven't found something in the books somewhere or online one way or the other.