Professor uploaded a recommendation meant for university 'A' to a different university 'B' There are 2 universities that I was applying to for a PhD program. Lets call them 'A' and 'B'. 
The professor, who was writing a recommendation for me, uploaded the recommendation, meant for university 'A', to university 'B' and the one meant for university 'B' to university 'A'.

This error was realized by the professor after I had already submitted the entire application, so I cannot remove the recommendation from my application and request a new one.

So my queries are:


Can I/Should I drop a mail to the graduate program stating such as happened and request a new link(or something along those lines)?
If the situation is not rectified, how does that reflect onto my application?
How it affects you is impossible to say, not knowing nearly enough. 

But it is really the professor's job to make the correction if it is needed as s/he made the error. You can send mail to the program(s) informing them of the error, of course, and they may, individually have some suggestions. But if the mail comes from the professor it would have more weight. The prof can include the proper letter in the mail, of course. 

But if radically different things were said about you (unlikely) it might have an effect. 

But ask your professor to provide the remedy.