iMac running on external SSD won't boot from it after upgrade to Catalina it seems a very similar problem has been reported. Nevertheless, its resolution is not clear and seems to be a matter of chance.
For clarity let me add the following background info:
For a couple of months I've been successfully running my 2013 iMac from an external SSD (for speed reasons). Initially, the SSD was a clone of my internal High Sierra Macintosh HD. Since then, program updates and content were added successfully to the SSD.
Today I performed a successful upgrade to Catalina. The system looked fine.
I then performed another recommended security + Safari update. When completed the Mac booted to the old High Sierra OS from the internal HDD!
I used the Startup preferences to set the external 10.15 OS X SSD as boot disk - no effect! still booting from HDD.
I tried rebooting into Recovery mode (cmd+R+power) to get access to the disk utility - no effect! the boot process form HDD continued regardless of that. This key combination worked fine when I cloned my HDD.
The same happened when I tried NVRAM reset (opt+cmd+R+P) - the boot process form HDD continued regardless of that.
I disconnected the SSD and tried again Recovery mode and NVRAM reset. This time the key combination commands seem to work, but even though I selected to boot from the SSD, which I reconnected, iMac simply booted from the internal HDD.
I made the above attempts following advices provided in various forums where success was reported on sporadic cases.
Is there any structured procedure (or another wild idea), to set it right? i.e. make iMac boot from the external Catalina SSD
Comments:

The external SSD is now displayed in Finder as two disks: Macintosh HD and Macintosh HD - data. This was probably created during the upgrade to Catalina. No idea why. (The internal HDD was never partitioned throughout all previous OS X upgrades).

If worse comes to worst, I have a Time Machine backup of the SSD prior to the Catalina upgrade. Not sure how to use it under the circumstances.


Thanx
UPDATE
I found that I can boot from my external SSD reliably as long as my external HDD for Time Machine is not connected. No need for the extensive process below - just power down, unplug external HDD, and my Mac can boot from the external SSD.
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I had the same issue and neither the SMC reset mentioned here or the NVRAM reset discussed here worked.
But the workaround I found to reliably boot off the SSD is:

Power off the computer
Disconnect the external SSD
Power on the computer (which will boot off the internal hard drive's OS)
Connect the external SSD
Login and shut down
Power on while holding the option key
Select the external ssd from the menu and the computer will boot off the SSD