That might be a clue - is your temporary folder /tmp configured that way? In other words, is it configured with the leading slash and then "tmp"? If so, that would mean that you are configuring it with an absolute path, which is the normal scenario at least for dedicated servers. If you are on shared hosting, this might cause an issue. My guess is that the issue is not actually with the module itself but with some other server-side config that is just tripping an error, which the uc_file.module doesn't quite understand how to parse.

The next question is for your "private" folder - I don't believe you need to chmod it 777 but that's what I have done with our downloads folder on our dedicated box. Try doing that and see if it changes anything. You might also try the different configs - absolute path vs. relative path. I believe both are acceptable, but absolute path is always recommended. It should be something like...

/tmp (temporary folder)
/var/www/vhosts/example.com/downloads_folder (folder alongside webroot)

Not sure if this is out of your expertise as well - but try tinkering with some settings there, so long as you don't mind changing things. I don't want you to break anything on the site (But there is (almost) always a fix!)