The way it currently works, I can see pets evolve to be progammically processed as just a graphical presentation, so there doesn't have to be delays.
In my comment, I was talking about the case where pet skills replace SMN player skills, i.e. they work just like any other job's skills, except that graphically it's your pet doing the animations, so there doesn't have to be any delays. You're not ordering the pet to do something, it's you who's casting the skill, except the game's graphics shows it as your pet doing the attack. For melee, since pets aren't targetable, they can literally stick to a target even better than any player can ever hope to do, by just doing the processing server side and updating the pet position to be shown on player screens.
But of course that means a complete overhaul of the inner mechanics of the job, so it was more of a musing rather than a serious suggestion.