A Different High Sierra/Time Machine Issue

I suggest using the following technique to "reconnect" to the backup disk. You will not lose or otherwise affect its existing backup history in any way.

Open Time Machine Preferences and de-select the Back Up Automatically checkbox. Make sure no backups are in progress, otherwise you'll have to wait for it to finish or terminate.

Then, click the "Select Disk..." button, select the backup disk, then Remove Disk, then Stop Using This Disk.

If you designated more than one backup destination, the "Select Disk..." button is replaced by an "Add or Remove Backup Disk..." option. You have to scroll through the list of backup disks for that option to become visible.

This will not affect any existing backups.

Then, Select Disk... again. Re-select the same one under Available Disks, then Use Disk. If the backup is on a network (TC or AEBS), you will be prompted for the password you provided for that disk in AirPort Utility.

"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.

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.

Although it is not a requirement, I recommend using more than just one backup drive, each one dedicated for Time Machine's exclusive use and nothing else—no other partitions or data, because a catastrophic failure of a drive with multiple partitions is likely to result in losing everything on the drive. More than one Time Machine backup drive will make sudden and catastrophic loss of all your Mac's data effectively impossible.

One other thought on your problem. I didn't notice the icons on your Desktop before, but it looks like you never formatted the drives with Apple's Disk Utility, but used the drive software that came with the drive. Notice the icon for your Macintosh HD? The icons for your external drives are different. Indicating you never formatted with Disk Utility.

If the drive software is not updated to work with High Sierra, you'll see issues like this. Check with the drive manufacturer to see if they have issued an update.

In the future when you buy a drive always format with Apple's Disk Utility. You don't have to buy a Mac drive. You can often save money and buy a PC drive. Format with Disk Utility and voilà it's a Mac drive formatted correctly for your Mac.

Is the external drive formatted as HFS+ or APFS. Time Machine does not work with APFS.

I suspect you'll need to wait on an update to fix or reformat the drive and start a new Time Machine backup. I would copy the current backup files to another drive before reformatting.

Please give it more time, perhaps overnight. During that time, leave "automatic" selected, and either leave your MBP connected to power (don't it shut down) or select "Enable Power Nap while on battery power)" in Energy Saver, or both.

Sierra introduced some changes in Time Machine, and High Sierra introduced even more. One of those changes appears to be that TM will not write a Time Machine backup to the backup drive until there is some amount of new data to be written, and examining its logs will reveal that circumstance to be logged as an "error". That "error" should not be taken literally and probably ought to be disregarded until more information can be determined. If Time Machine's Preferences do not indicate the creation of a new backup for another day or longer, then please write back for a suggestion to extract those logs so that they can be examined.

I recommend leaving "back up automatically" selected. De-select it only for justifiable reasons. You may have some, but I can't think of any, and leaving that option de-selected will just complicate troubleshooting. Leave it on for a while and see what happens.

To my knowledge initiating a manual backup should always work. If the backup volumes are mounted (shown on the Desktop) when initiating a manual backup, but TM complains they are not, then something else is wrong.

I can't explain the inability to read or write from the USB hard disk. It strikes me as a premature conclusion to draw though. It would help if you had another TM backup drive, but I realize that might not be an immediately available option for you.

I have generally found it necessary to only run the TM backup once every week or every two weeks. For that reason, the "manual" option works best for me.

It really, truly, does not matter, and the reason I recommend leaving it on "automatic" is to advance troubleshooting since TM's automatic backups are the problem you're encountering.

"Automatic" is how I use all of mine, what I'm familiar with, and the configuration for which I can best offer assistance. If the backup drive is not available when TM would like to back up, that's a normal and expected part of its design. TM won't complain until ten days elapses without a backup (for whatever reason). If TM isn't allowed to work "automatically" every day, initiating a manual backup can take a very long time.

Restart your MBP with the backup drive connected, select "automatic", and give it a day or two. The lack of a "Back Up Now" is strange though, and may indicate the need to reinstall macOS. I know of one report on this site that required it for a similar problem (not backing up automatically). Reinstalling macOS is nondestructive and simple enough, and might be worth trying.

You can either wait to see if a future update fixes the problem, or you can erase your backup volume then start a new Time Machine backup.

If you erase, you'll loose all your old backups.

Open Disk Utility in Applications/Utilities.

Select the drive/volume.

This shows a drive formatted as HFS+ Mac OS Extended Journaled

This shows a drive formatted as APFS

You do not want to format your internal drive as APFS at this time. The new format is designed for SSD drives. Currently, APFS does not work with Fusion Drives and hard drives (HHD).

Great! Thanks for the update.

Temporarily disconnecting the backup drive is an important detail that I neglected to include in my instructions. Your comments are very helpful and I will make it a point to add that step in the future. Many thanks.