Thanks for looking at this.
I tried that in my original post, but revisited it and got a bit further.
The Install Composer button downloads a file which it renames composer-setup.php, so I manually downloaded that and ran it from the Tiki directory.
That gave this output:
Notice: Undefined variable: argv in C:\inetpub\wwwroot\TikiWiki\tiki-19.0\composer-setup.php on line 14
All settings correct for using Composer Unable to create Composer home directory "C:/Windows/system32/config/systemprofile/AppData/Roaming/Composer": mkdir(): File exists
I'm not a great PHP programmer, but the Notice seems to be triggered because a command line argument wasn't supplied, and I'm not sure that one is needed.
Then it passes through all of the checks OK
But then the error at mkdir is strange because, contrary to what it says, there isn't a Composer folder in the Roaming directory.