What is the command to execute after manually editing the composer.json file? I want to build a site broadly based on an exiting site and was thinking that the best way to do this would be to copy and paste required modules from the existing site composer.json and paste them into the new composer.json. What would be the correct command to download the modules via Composer once the file has been amended?

I think composer update is the command, but I don't know if it also updates dependencies that should not be updated.
You are better off calling composer require some/library. This will update the files accordingly and install the library as well.
That said, you can run composer install after manually editing composer.json.