Get an old PC and a bunch of PCI-USB cards (remember that USB bandwidth is shared, so the fewer devices you have per controller, the faster it'll be). Under Linux it'd be very easy to write a script that would detect USB insertion, format, and copy files - maybe an hour's work. If you really want to do some electronics, interface a bunch of red/green LEDs that indicate the status of the copy process for each flash drive so that people can just unplug and replace drives when the light goes green.
You could probably do it with a microcontroller, but unless your time is worth absolutely nothing, it'd be more than £700 in labour. The advantage to using a PC with a proper operating system is that it can be agnostic to the size of the flash drive you're connecting - it'd be a huge amount of work to write firmware to format a FAT32 file system and copy file-by-file.