At first I have to say, that everything was working before on bullseye with old node red 1.X. After I started the upgrade script it didn't launch as it said that "Nodejs 10 too old and new version not specified - exiting".
So I launched the script with the command line option to force install nodejs 16.
During the installing of this it screwed up the file permissions in /usr/bin.
All files were "owned" by the uid 1001 which does not exist at all on my system (user pi has uid 1000).