I applied for both the Marriott Chase card and the Delta Gold AMEX card on the same day -- Thurs. April 28.
Chase did NOT approve immediately.
I didn't find out I was accepted until I called Chase on Mon, April 2
Card just arrived today -- Mon, April 9 (9 business/ 11 calendar days later)
(If I hadn't called, I would not have know I was even approved until the card landed in my mailbox.)
AMEX approved immediately and gave me a temp card# to book a flight with on the spot.
AMEX sent an email on Sat, April 30 saying the card had shipped.
AMEX card arrived Mon, May 2 (2 business /4 calendar days later)
So far I give Delta AMEX the win, hands down --- no love for Chase
Chase did give me a larger credit line, but I really don't care about the credit limit.
Chase is odd with the shipping of their cobranded cards. they quote times of up to two weeks just for processing, due to the "complicated" linking of the cards to the marriott account. Once the card populates in your chase online account (if you already have one) that means your card number has been generated, and your card will ship. You can call them the day you notice this, and ask them to send it expedited, which will be UPS Next Day Air Saver (no saturday delivery). If this happens on a friday, you'll need to specifically request saturday delivery, which is UPS Next Day Air (Standard).
These requests need to be made by about 4PM EST at the latest, for shipping to occur on the day you request it, as they ship out at approximately 7PM EST.