I don't know how long it takes because I ordered mine on line about a month ago and called last week to check on it because it hadn't arrived - she had no record of the request, so said she'd send one out to me.
FYI - when I ordered my first card last year, I only put the required $20 on it. Once you move $$ over to the reward card it carries an expiration date of about one year (not the 5 or so years you have if you leave it in your account). Just in case something happens to my trip I like risking only the $20 - then I call the 800# and have the rest moved over to the card once I'm there - it's transferred immediately. Actually, I had a large reward balance and only moved over what I had spent the night before check-out.
I was able to use the rewards listed on my current statement before I had actually paid the bill (we went mid-month - billing period ends on the 4th for us and is not due until the 30th)- that surprised me, but it was a nice surprise!
Long story short, I would order my card with $20 now (I already ordered mine for June) and transfer the rest once I reached the world.