Give the gift of Drupal. All merchandise is 50% off through 2016.
Is there a way to show the url of the uploaded file in the edit form, like the core module Upload?
...In drupal 5 version, the file's url was also visible in the edit form.
FileField won't be changed at this point since it's been moved into core, but you can accomplish this by theming your FileFields. I think overriding theme_filefield_widget_item() in your template.php file would probably be the easiest approach.
As an alternative, if you're wanting to copy/paste that URL into a textarea, you'd probably find Insert to be a better alternative to copy/paste.
Drupal is a registered trademark of Dries Buytaert.