Change the $file_list from a string to an array.
$aFiles = array();
while (false !== ($file = readdir($handle)))
$aFiles[$file] = filectime('./content/uploaded' . $file);
The files will now be sorted by date created. From here you can create the string and the date using a foreach on the $aFiles instead. The key is your filename (which will be unique of course), and the value is the create time.
There is a way to do this with Iterators as well, although (as much as I like objects) they are a little difficult to do properly since PHP doesn't have a default comparable method on its objects.
Reading the file's name won't issue any type of touch command so the modification date will not change.