My most recent petition worked like this (when I was using bandolier my weapons would disappear. I would summon shissar fangs and they would disappear. Then, randomly, after clicking around, some stuff would come back, but then disappear again). I petitioned on a Sunday I logged out. I logged in Monday Issue fixed itself (I had 8 stacks of shissar fangs on my cursor when I logged in) I emailed in response to the auto email for the petition saying problem is fixed. A week later I got a response saying "thanks for letting us know" blah blah So, a week was my experience.
I also think the type of petition will determine your wait time. For example things that are related to finances probably take priority to bugs and bugs take priority to broken quests and so forth. Basically depending on the category you select probably filters things into different levels of priority.
Just took me about 5 days to get a response asking me for information to confirm I was account owner, then another 2 for the reply ok problem fixed.
The problem category is super relevant. If they can forward it to a game GM, those guys sometimes bust out 100 solutions in an hour. Like, bounce a bugged zone or whatever. Suspend someone with logs of racist threats in #general? Happens super fast. If the problem takes serious digging, though, that GM's going to prioritize the 100 easy probs ahead of spending an hour on your one. If it's acct level action, the CS folks have to do verification stuff. Lots of layers.
Actually filled one out about 10:30 last night EST and had a response around 8:30 this morning. Just not a response I wanted. Categorized as a deleted item, but it was some funky bug where I didn't get the money for an item that sold in bazaar, and that was the closest category I could find.
I did a weekend petition a couple of months ago and a GM resolved it within 2 hours. Other times it was taken care of the next day.