photos stuck updating library which is is on external drive connected to imac M1

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 

• Make a small test Library on your internal drive with a few pictures in it, set it as the System Library, and see if it begins to sync. You'll need to stop it before it downloads your full iCloud Photos!
• 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. 

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.

Let us know what happens…

Just to add to Yer_Man's point:

If this drive is in a an incompatible format, stop running Photos with it immediately!  A Photos Library can sit on an incompatible drive, but running it may corrupt the database.

Older OSs didn't block non-apple formats, and corruptions just slowly oozed into Libraries until they would suddenly go crazy. So this is certainly the first thing that needs to be checked.

What format is the external disk? It need to be formatted Mac OS X Extended, Journaled or apfs.

How is your external drive connected to your Mac? It needs to be connected with a cable, not a network connection.

And how much free storage is on the external drive? Upgrading the library to Tahoe may hang, if there is not enough free storage on the external drive for Photos to upgrade the library.

Just so I am clear:

IN the title of the post you say that Photos is "stuck updating the library" but in the actual text you say it's "stuck on syncing with iCloud"

Can you clarify? Also, what happened when you launched Photos after upgrading to the new OS?

BaysiderAus wrote: … phone support and we created a newlibrary

Great! Sounds like our step 3!

on the same external drive initially optimising the mac storage but when downloaded completely will transfer to storage on hard drive.

I wonder why they suggested "Optimize Storage" on your external drive. Did they say how you would know that your pictures had finished downloading originals after you switch?