Users can only use their nickname if it is entered in their profile. To see if it is entered in their profile, you'll go to:
  1. Core > Users/Access > Profile
  2. Search the user
  3. Edit Personal Information
If the First Name field or Nickname field have the name they are trying to enter, then they will be able to go through the Forgot Login process. If it is not there, they will receive the message:
We are unable to process your request. Please contact your system administrator for assistance.