What is the most logical course of action if your P.I does not write you a letter of recommendation in time even though s/he said that s/he would?
I'd say your best course of action is to notify the institution that you are applying to that there has been a delay, and to find someone else who can write you a recommendation within a week's time. The admissions department is likely to have a little bit of grace period for you to work with, but probably not more than a few working days. That said, they may not accept the late recommendation, but it is worth trying.