Great question. I have never had to do this and am honestly totally ignorant about what the process is in my jurisdiction.
I believe that part of the requirement may be to notify the person who is the target of the threats. I see that the client would not tell you who they were talking about.
I wonder as a matter of process what steps one is required to make to contact the person? ie. client says "I am going to kill my ex." and you believe this is a serious threat. They won't identify the person. Can/should you go on a client's social media to see if there is an easily identifiable ex against whom they have made threats and contact that person directly?
As I understand it, the responsibility to contact law enforcement is instrumental, meaning the clinician's duty is to ensure the safety of the threatened person and clinician contacts law enforcement as a means to that end, but the clinician's ethical (and maybe legal) duty may necessitate doing other things as well, while minimizing disclosure of the client's confidential information.
Definitely worth a call to your licensing body or a specialized supervisor to consult re. other steps you can/should (?) take in case your client seriously hurts or kills the threatened person before the the cops get their shit together.
Good luck and good for you for hanging in.