2 days at the very least depending on time of day ordered (if I remember correctly, the cut off time was early, around 3pm for an order to ship same day), then anything after that (a week at most with items in stock). The most likely scenario is that Apple is out of stock on said part which could vary extremely depending on what computer and what part. I've seen a week, a few weeks, and a couple months on ETA for out of stock parts on old machines.
As a authorized center, we couldn't do anything to speed up the process as it was solely in Apple's hands, unless we lied and claimed to be the customer (which got old really fast). The best we could do would be to give the customer the repair order number and Apple Care number, to query their repair and the ETA on the part, and then be nice but firm about a much delayed ship time. Most of the time Apple would miraculously find a part to ship the next day after speaking to a customer, and some other rare occasions (with really nice people, and ridiculously long ETA's) the computer would be replaced if no part was due for months.
