That's ok.
First, de-select "Back Up Automatically" to make sure no backups are in progress. Then, go ahead and click Remove Disk and confirm the dialog box that appears next: Stop Using This Disk.
That action does not affect its existing backups.
Then, physically disconnect the drive from that Mac and connect it again. Wait for its icon to appear on the Desktop, assuming you selected External Disks in the Finder's Preferences.
Then, Select Disk... again in Time Machine's Preferences. Re-select the same one under Available Disks, then Use Disk.
"Waiting to complete first backup" will appear, which means it won't do anything until the next scheduled backup. Despite the implication of that message, it does not mean it needs to create a completely new backup, as if it's creating a brand new one. Your existing backups will still be available.
If you don't want to wait that long, select Back Up Now.
A Notification may appear if your source volume is encrypted and the backup volume is not. That's normal, although you really ought to consider using encryption. Selecting "Encrypt backups" will require erasing the existing ones first so that's a concern to be addressed later.
Re-select Back Up Automatically. The "Preparing Backup..." status message will remain for a long time. Eventually, "Backing up xxx of yyy" will appear, but those values will not be accurate, nor will the "Estimated time remaining" in Time Machine's Preferences. Just ignore it. It may take a few hours even if your Mac is not allowed to sleep. Time Machine will also pause or become slow if you use your Mac for tasks it considers a higher priority.
If the above does not occur for whatever reason, consider the possibility the backup disk has failed, perhaps catastrophically, and it might not even appear in the Finder. You can determine that by using Disk Utility.
