Can I change my recommendation letter provider?
Contact the admissions department, explain that the professor is "caught in some private matters" and hasn't been able to do the letter as a result.  Ask to switch the reference to a different prof.  

In my experience, you're much better off being certain that you don't miss the admissions timelines due to a delayed reference letter than the source of the letter mattering very much.  That being said, I'm not familiar with the specifics of UIUC, so my experience may not be relevant.