I called for this last week. Start dialing 6:55 am, then redial til you get through. I used 407-WDW-DINE.
If you hear a male voice welcome you to WDW Dining, he will then say, "Our office is closed..." Hang up.
Once the female voice answers, you will have to punch in some numbers to get to the actual reservation agent.
First prompt: "If you are calling to make a priority seating for any location other than Victoria and Albert's, PRESS 1."
Second prompt: "If you are calling from the Orlando area to make a PS within the next 7 days...press 1" (doesn't apply to you)
If you are calling to make a priority seating for an upcoming trip, PRESS 2."
Third prompt: "If you visited the WDW resort before, press 1" (don't do it, this just takes you to another prompt). Instead, "If not, PRESS 2." This will take you to the actual agent.
I pressed the corresponding numbers as soon as each of the prompts began. I would also suggest that you have a script in front of you: "Party of 5, anytime for Cinderella's Royal Table on July 6. "
Hope this helps.