Do they use the due date or the grace period date?
Two important dates for mortgage payments. The late fee kicks in at the end of the 15 day grace period, and at that point additional charges are incurred. However, lenders do not report mortgages as late to credit bureaus unless they are not paid in the month due (ie: paying March 1st payment April 3rd). While late fees are not fun, they are far less damaging than a 30 day late on your credit, which has the potential to dramatically affect your credit and ability to refinance or qualify for a new mortgage.
For the answer to this question for a specific loan you would need to refer to the loan agreement, but in general:
Most mortgage loans are due on the 1st day of the month. This payment includes interest accrued from the previous month, the required amount of principal, plus any amounts required for taxes and insurance.
If a payment is not received by the 1st day of the month it is late. Most loans have a 15 day grace period. After the grace period a late fee is added to the amount due. Usually 4% or 5%.
If a payment is not received by the lender in the month in which it is due, the loan is in default and needs to be brought current to avoid collection efforts. Additionally, a loan payment not made in the month due, i.e. a loan payment more than 30 days late, are often reported to credit bureaus as a late pay. If not paid before the next payment becomes due then the payment is reported as a 60 day late, etc.