I’ve had the same problem except that all my letters writers did the same thing to be (I got my letters in last week, still waiting on 1 actually...). The email that worked the best was to the letter writer saying:
Dr. Chairman,
Thank you again for agreeing to write me a letter of recommendation. I recently interviewed at --, and was questioned about when my LORs would be submitted. I have another interview (next week, this week, etc...) and feel that a letter of recommendation from you, reflecting my time at XYZ, would be very beneficial. If it is at all possible for you to submit the letter by (above time frame) I would be immensely appreciative. If there is anything I can do to expedite the process please do not hesitate to let me know.
Thank you again for all your help,
Neurotic