Ah, that's probably because their initiative is already spent after you had yours, I think, or something like that. I'm not sure of the exact game mechanics there, but it seems to have to do with the initiative value which every enemy, and player characters, do have. You can look it up for your party characters with the Cosmic Forge editor through Save Files Viewer, the value raises with each level. If yours is higher than that of the enemies, you begin the round first.
The enemies seem to be able to act only in the next round then. It seems to be random, so sometimes they also act in the same round you awoke them through melee hit or spell.
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