Okay, I downloaded the EaseUS Data Recovery Wizard trial version. Unfortunately, the application does not seem to be able to "see" the hard drive, in order to conduct any kind of partition recovery. I'll assume that's because there is no formatting or other table information available for the application to use - to even begin the partition recovery.
Clearly, I've deleted the existing partition, but I don't want to do anything that makes the situation worse. I wondered whether the next step is to conduct a Quick Format. Doubly unfortunate, is that my external hard drive enclosure can power-up and access the drive, but the drive doesn't show up via My Computer or Windows Explorer. Obviously, I can toss the drive into the laptop, and run a format (Quick Format) procedure on the drive. That should at least lay down formatting - which should allow the drive to be accessed ("seen").
Any recommendations on how to proceed?