Make sure you have set the external drive as the default in System Preferences. See the apple support page I linked to see how to set the default boot option.
What I recommend you do is first re-install Catalina on the disk. Re-installing DOES NOT ERASE YOUR MAC. It just replaces system files and removes security updates released after the release of 10.15.7. To do that download the full Catalina installer from here: https://apps.apple.com/us/app/macos-catalina/id1466841314?mt=12
Then launch the installer and select the external drive that you are booting Catalina from - MAKE SURE you select the correct disk - or it will upgrade Mojave on your internal disk to Catalina. It will install and restart your computer. Then try updating again by following the below steps:
- Download the update (either from the package or Software Update in Sys Prefs)
- Wait for it to install and then asks you to restart
- DO NOT restart - shut down computer MANUALLY and then turn it back on
- Hold option key to MANUALLY go to the external disk where the update needs to be
- See if you get that White Apple logo like you do when you startup, with a little bar that says some minutes remaining. EXAMPLE:

If you don't get the bar and logo described above, that means there is something wrong and the update isn't even working when downloading/installing before the restart. If you get the logo and status bar with the time remaining as described in step 5, and it stills shows in Software Update that the update needs to be installed, then it might just be a weird bug and takes a while to recognize that the update was successfully installed.
Let me know if there is any problem EVEN after doing what I described.
Hope this helps.
Cheers, Jack