Really quick. Like a week or two? It took longer to get his birth certificate because I had to call the county and do a few things to have them send it to me.
You can check the estimated wait time for your area online. In my area, the wait was 8-10 weeks each time. When DD1's didn't a arrive in that timeframe, I had to apply for a replacement. DD2's arrived well before the estimated time.
Ours hadn't show up at 10w, and I ended up calling our local office. Apparently SS had issued a number but never mailed the card, so I had to go to the office in person to apply for a "replacement card." It was a PITA.