Thanks everyone for the info. I tried a new micro sd card, and that didn't help. I then setup Octoprint on an rpi3, plugged that into the duet via USB, and my file upload to the rpi3 is extremely fast.
Conclusion: there is something either by the Universities network or the Duet's wifi firmware causing issues with uploading to the Duet.
Now I have Octoprint setup, which I wanted anyways for the webcam streaming. The only problem is I liked DWC more than the Octoprint interface.
Is there any method of transferring files from the RPI3 running Octoprint to the Duet over USB? That'd be clunky, but if possible then I could use the DWC for everything but uploading files.
Also out of curiosity, are there plans to sell the Duet Ethernet's daughter board as a standalone? I've seen mention of that on the forums, but I couldn't find anything in the shop. I might just buy the W5500 and make my own circuit in the meantime.