Waiting for approval from professor to write letter of recommendation
Rather than contact the other university, make sure that the professor in question is aware of the time situation. But, if everything else is in place for your application, it seems unlikely that the university would withdraw the position for any single piece of information that isn't completely in your control.
You might want to consider asking an intermediary, such as another professor or the department head, to make contact with the professor to provide a bit of a nudge.
Let the other university know, perhaps, that you have requested the letter from Prof XYXXY.