The only workaround I found is to include the badge health timestamp, but it's not a real-time info, but is the closest time I can get.
Is there any other solution?
Concerning the filename...
If you download the CSV or the PDF from the UI or send the files by mail, the format name is month-day-year Reportname.
When you upload the scheduled report directly on a Network share... the file name is different!
It will be: month-day-year time_stamp Reportname.