Getting your export
My downloaded files are only a few hundred bytes. What happened?
The download link from Snapchat had expired by the time you tapped it. Your phone then saves an error message under the filename instead of the file. It looks normal, with the right icon and everything, but weighs a few hundred bytes instead of hundreds of megabytes.
The files did download. What downloaded is an error message.
When you tap an expired link, Snapchat's storage does not answer by saying the page is missing. It answers with a small document explaining that the link has expired, and your phone has no way of knowing that this is not what it asked for. It saves that answer under the filename it was expecting, and reports the download as finished.
The result sits in your files with the right name, the right timestamp and the right icon. iOS picks the icon from the file extension and not from the contents, so nothing gives it away.
The only visible clue is the size. A real part of an export weighs hundreds of megabytes. If it says a few hundred bytes, the file is empty however normal it looks.
Here is how to get them anyway: open Snapchat, tap your profile, then the gear icon, and scroll down to My Data. Download from there rather than from the old email. The page creates fresh links when you tap, so the same export can usually be fetched again without ordering anything. If the export is no longer listed at all, ask for a new one. It is free, and you can do it as many times as you need.
Download as soon as the email arrives next time. The links do not last forever, and an export left overnight has often already gone stale.
Your memories are not gone. It is only the download that failed.