Photo library on external drive not syncing with iCloud or other devices

You need to look over all the posts including Servant of Cats. Your screenshot does not show the library is on the external drive. Please close the Photos app, navigate to the external drive and double click where the library has moved to. Then give iCloud time (a few hours) to sync. If it is still not syncing then you need to give AppleCare a phone call for some hands on assistance.

AppleCare Contact Info

Once the library is on the External Hard Disk, did you double click the library in its new location? You would do this with the Photos app closed. When you double click this launches Photos and "tells" the Photos app where the library is now located. After that Photos will work identically to how it was already installed. You can verify the new location by looking Photos preferences.

If it does not then you simply did something wrong, so you will need to go back to Apple's instructions ( Move your Photos library to save space on your Mac ) and find your error and correct it.

Have you checked if the external Photos Library is the System Photo library? Only the System Photo Library can sync with iCloud.

Congratulations!

Library Repair is well documented, but sometimes we here get focused on a problem and neglect the more general checks. And rebuilding the Library can take a long time for re-syncing for iCloud Photos users, so we're careful in recommending it.

Just for your interest, here is my suggestions for when I have no idea what's going on:

To narrow things down, try these steps:

• Restart the computer (of course)
• Re-start in Safe Mode. This bypasses certain potentially disruptive processes, and it often helps. Safe Mode is different for different computers, so see this: 

           Start up your Mac in safe mode - Apple Support 

• Use another library if you have one, or make a small test Library with a few pictures in it and see if the same thing happens. You can create a new Library by closing Photos and then option-clicking on the Photos icon.
• Make a new user and see if the same problem recurs with it.  This is a bit harder, because a new user can't access a different user's stuff. You would need to move your Library out of your own user's folder up to the general "Users" folder. 
• Rebuild your Photos Library--close Photos and re-launch by option-command-clicking the app icon. If you use iCloud Photos, this may force a re-sync that could take days to finish.

The idea is to find out if the problem is with the Library, with your account, or with caches and login items, and stuff like that. One of these steps may fix the problem, but their failure to work also helps figure out what's happening.

artshot wrote:

I recently loaded about 30K photos into the Photo app on my iMac desktop, Photo Library is about 1T.

Obviously too big to be stored on my computer's local drive, so I followed Apple's instructions to transfer the library to an external drive which is permanently attached to my iMac.

How is that external drive partitioned and formatted...?

Did you move part or all of the Photos Library to the external drive...?

Here's what I can't understand: my desktop's Photo app has all the images available in the Albums that I setup and all changes are updated on that machine if I make any changes or additions, but the Photos apps on my IOS devices as well as on my laptop do not show these Albums. Why not? On iCloud the images that I loaded into Photo app while the Library were still on my computer's internal HD are available, but any that were loaded after the Library was moved to the external drive are not. If the system has been told where the Photo Library is to be found, why doesn't the system continue to update iCloud so that the photos are viewable on all my devices. My iMac system settings are to Optimize storage so that the originals are not stored locally but in iCloud. Nowhere have I seen it documented that if the iPhoto Library is stored on an external drive your Photo iCloud Library is not updated.

I'm not an iCloud expert, but it sounds like you need to keep the Photos Library on the external drive stored locally and in iCloud for sharing with your other devices.

If you hold down the Option key while launching Photos, you will get a dialog that lets you switch between Photos libraries, and see where each of those libraries is stored.

artshot wrote: I must admit to some degree of frustration.

Hold on! We here are just users like you, and some of us hang around here from time to time in case our experience may help someone. It's amazing that we can help at all without seeing your machine and relying on what you tell us-- our AI Mind Reading App is still in development! Much of this depends on what's most probable, which we get from the questions that are asked, here. You are always welcome to take your device to a computer service and pay them for their time.