This is a very important query.
First, as a consumer, you should verify at closing which method will be used to payoff the mortgage. Is the settlement agent wiring funds? If they are sending a check, is it going out via courier or US mail?
I don't trust payoffs to the regular mail. Interest is accruing and do YOU really want to wait and how long would it take to find that something was lost?
The funds should go out from the settlement agent's office the day of closing or the next business day. The mortgage lender receiving the payoff may have a mid-day cut off time. Many do not post after 2pm.
So, with all that in mind, you should expect a mortgage payoff to post within 2 or 3 days after closing. Beyond that, consider it a problem and make a phone call.