I have an external 2TB USB hard drive that is attached to a home LAN router, which is used to store file back-ups and for media files that can be accessed by any computer.

It appears that that the existing NTFS file system is corrupted, so I started the re-format process, which I see will take 14 hours to complete! (I am using the erase option). Anyway, while "watching the grass grow"; it occurred to me that it might really be a better idea to simply re-format the drive as EXT4.

All the computers are running Linux (Ubuntu) anyway, so there is a minimal need for a NTFS formated drive. Any downside? Your thoughts?

Even with a 2 TB disk formatting to NTFS should only need a few seconds, unless you forgot the quick format option in your command.
If all computers this disk will be connected to run Linux then there is no downside to using Linux specific filesystems.

Correct, I forgot to mention that I am using the erase option, which I knew would take longer, but not that long. The total time was not too much of a concern as we have to leave for several hours anyway. But, the more I think about it, the more EXT4 appears as the better alternative. When we return, I will cancel the existing format operation and start over.

Success. The major gremlin: the router only supported the Linux file system up to EXT3. So I had to remove all the back-up files, re-format to EXT3 and then reload the back-up files.