Moving the site should not impact the ability to upload (assuming it worked before the site was moved) as long as folder permissions are correct. You might try clearing the browser cache in case anything got cached in your browser with the old urls.
It really does not sound like the App_Data folder (or the sub folder used by NeatUpload) is writable. If it is not writable it will cause this problem because the temporary file will be located under Windows temp folder instead of App_Data and this makes permissions funky after it moves the file out of the temp location. You might watch the app_data folder and see if any file is written there during upload.
Hope it helps,