A BTO isn't entered into the system the day that you order it. It's normally not even entered the same day your dealer orders it.
BTO's are collected and entered into the system, en masse, once a week. When they're entered, they're not started right away. I'm not sure what the time frame is now but, a while back, the start date of a guitar could be as far out as a month from the date that the BTO was entered into the production schedule. I believe they've pared that down a bit.
Normally, the Rep will give the store the expected ship date if the dealer asks for it. It is absolutely the dealer's responsibility to get, and pass on, that information.
One thing I would stress is for you, as the end user, not call Taylor for that information. Ask your dealer to do that...