How the dates are counted
"Business days" in Publication 1281 means days other than Saturdays, Sundays and legal holidays. This calculator uses the eleven federal holidays (New Year's Day, Martin Luther King Jr. Day, Washington's Birthday, Memorial Day, Juneteenth, Independence Day, Labor Day, Columbus Day, Veterans Day, Thanksgiving, Christmas) with the federal observed-day rule when a holiday falls on a weekend. Day 1 is the first business day after the start date.
The 15-day deadline runs from the later of the notice date and the date you received the notice, so keep the envelope or log the receipt. The backup-withholding deadline runs from the notice date regardless of receipt. The payee's response window runs from the day you mailed the B-Notice, which is why mailing early buys you nothing on withholding but does give the vendor more time before it starts.
Missing the 15-day mailing deadline doesn't create a penalty by itself, but it removes the B-Notice as a timely solicitation, which is what the reasonable-cause waiver depends on. Missing the 30-day withholding deadline creates direct liability under IRC 3406 for the tax you should have withheld.
Full procedure: CP2100 and the B-Notice, step by step. Letters: templates page.