USB External Drive Not Mounting?

Did you try repairing/First Aid on the disk?

Is this a rotating disk?

How old? The disk may have failed.

Can you hear/feel if the disk is rotating if it is rotating disk?

Time Machine disks are Read only except by Time Machine starting with those created with recent macOS versions.

when repair takes a long time, it can indicate that a drive is accumulating errors, and may be dead soon. You should obtain an additional drive and start an additional backup set. Just set the new drive as an additional time machine destination, and time machine will alternate -- every other backup to every other drive.

Since the new drive is a stand-alone backup set, the first backup to it will take all afternoon.

You are free to remove the old drive at any time.