Since there doesn't appear to be a good solution to this one yet, how about a bad one ;-)

Your script could record the sizes (etc.) of all the files in the upload directory some time before the half hour mark, then go to sleep. When the time comes for doing the copying, the script only works on the files whose sizes are unchanged from the previous snapshot. I'm guessing that 5 minutes ought to do, but if your senders are as iffy as you say, maybe you want to take several snapshots.

One other idea that you probably considered and discarded, but placed here just in case: If there is any consistency to the file structure, can you detect whether they are complete by the content in any way? I'd like to suggest you get the submitters to put a standard end line of some kind but it sounds like that won't fly.