Here is one possibility, but not sure it applies to you. If an external drive is failing or the file system is badly damaged, when it is connected, the Mac may try to run fsck (file system check) in the background, having detected problems, before mounting the drive. In principle, this can, if allowed to run to completion, fix the file system problems. However with a several TB drive, especially one that has problems, it may take some hours to complete, or even more than a day. Activity Monitor may show fsck (or one of the variants on fsck, such as fsck_apfs or fsck_hfs) running in the background. When this happens, the disk is kept busy but does not actually mount until fsck completes. You can tell this is happening by running Disk Utility which should show the drive but the volume(s) not yet mounted (greyed out). In severe cases, Disk Utility may freeze, which means the drive has failed (or is close to failing).
In my experience, once this starts to happen, the drive has reached a state of failure that is so severe as to be unrecoverable. You may have to let this proceed overnight until the drive actually mounts. Sometimes then the drive is very slow or even freezes the Mac, more signs pointing to a failed device.
Before condemning the drive (it might be fine, there could be some other problem), I would shut down the Mac, disconnect all peripherals, then start up the Mac in Safe Mode. That may take 10-20 minutes, fsck is running in the background as well here but this time on your internal drive. Once that process completes, reboot normally. Then when booted up, connect the external drive and watch what happens. It might be fine, or you may see symptoms like I describe above.
In the meantime, get a new backup drive and tell Time Machine to continue to back up to both drives, it will try to alternate but will at least do backups on the new one, while you figure out the issue with the old external drive.
Another tell tale sign of a failed drive -- it won't mount on a different Mac, same delays, slowdowns, not mounting etc. but with disk activity, etc.