Time Machine doesn't show backups

I have experienced the following symptoms you described:

But now, Time Machine shows no backups at all.

It shows oldest backup: none. Newest backup: none.

Though not supported by Apple, I was able to reconnect to my old Time Machine backup (sparse disk image bundle) in about 5 minutes.

A big caveat for you, cleblanc65 is that you are adding an external disk, whereas, I only needed time machine to see the old backup in the internal Time Capsule disk again

Short Answer: Replace "com.apple.TimeMachine.plist" on your computer with the newest version from your old backup.

Long description--I put together a detailed step-by-step, while it is fresh in my mind, if that helps. Mileage may vary.

GETTING STARTED

1. Turn off (uncheck) Time Machine automatic backups in System Preferences. Do any local data transfers now.
2. Connect to your Time Capsule, etc. on your LAN via ethernet or wifi. Do not connect directly to the Time Capsule.
3. Use AirPort Utility to prep disks, but do not delete or erase any files you need without a backup (step 1).
4. Name your backup volume the same as the previous one (rename any duplicate volume names first, if any).
5. Note: Configure Finder to show invisible files if you can't see files in the following steps.
6. Go to "Macintosh HD/Library/Preferences/"; rename "com.apple.TimeMachine.plist" to "com.apple.TimeMachine.bkp.plist"
7. Give Finder permission to change the filename. Close all windows, eject mounted disks, network shares, etc., and restart.

OBTAINING LAST WORKING BACKUP'S PLIST FILE

1. Mount your Time Capsule shared disks via Finder sidebar "Locations" as 'Time Capsule' or 'Network' icons.
2. After mounting, looks for a file named after the computer when that backup was active.
3. In list view, your backup will have a file type of "sparse disk image bundle."
4. Double-click to open this disk image. Shortly, you should see a "Time Machine Backups" icon appear on the desktop.
5. Open the "Time Machine Backups", folder, then open the folder named after your [old] computer
6. Open the "backups.backup" folder. At any point a folder seems empty, take a break, it's just sorting a few million files...
7. Open the backup folder with the most recent date in its name, then navigate to "Macintosh HD/Library/Preferences/"
8. If "com.apple.TimeMachine.plist" is not in this backup's Preferences folder, try the previous date/time, and so on...
9. Copy the the latest version of "com.apple.TimeMachine.plist" from your old backup to your computer (Getting Started, step 6)
10. If macOS has already created a new copy of "com.apple.TimeMachine.plist," just replace it (you "bkp'd" it previously).
11. Restart again. Open Time Machine preferences and Select Disk... Choose your disk and you should see the correct Oldest backup and Newest backup information!
12. If you do not like the results, you may restore your original backup settings by turning off Automatic Backups, deleting the new "com.apple.TimeMachine.plist" from your computer, renaming the ("com.apple.TimeMachine.bkp.plist") back to "com.apple.TimeMachine.plist". Restart.

Ok, this worked like magic for me, but everyone's situation is different, so I went into high detail. Hope it helps.

I did want to mention that some people have changed out the internal TC hard disk with a larger one. I have not done this, but there is plenty info out there on this topic. Aside from voiding the TC warranty, this might help with volume management.

Inherit a Backup

If the above doesn't work, start with A4 in the 1st linked article.

Time Machine Troubleshooting second source Time Machine Troubleshooting