First thing suggested in check in the Applications folder and look for new application called " Install Monterey ". If it exists Drag and Drop to Trash and empty the trash.
If it does not appear in Applications folder - then reasonable to conclude it has not downloaded. The why is important but there is a way around this.
In the Terminal Application use this command without the quotation mark " softwareupdate --list-full-installers " One can copy and paste the command into Terminal.
It will only list all available Full Installers. If the output shows Monterey 12.1 - Good. It is available.
Next command, again without quotation marks " softwareupdate --fetch-full-installer --full-installer-version 12.1 " Copy and paste.
This will pull the full installer of Monterey 12.1 - it is about 12.17 GB in size.
It will only download and install a New Application called " Install Monterey " and will Not Launch.
Suggestion - make a Time Machine Backup before moving forwards.
Also, suggest is to copy and paste the " Install Monterey " to an good external Drive for future purposes like making a Bootable Installer
Now, suggestion ( yes full of suggestions lol ) restart the computer in Safe Mode .
Once logged into your Admin Account - this is when to Launch the Install Monterey located in Applications folder and follow the prompts. There will be several Automatic restarts and hopefully the final restart where Setup Assist will be presented.