Are you able to ping each other?
Can you pinpoint which ports WebRTC tries to talk on? (Look at ICE candidates SDP.)
Can you exchange data on one of those UDP ports? Can you exchange data on any UDP port?
Does your friend have a public IP address, and could he temporarily expose his/her computer via a public IP address instead of NAT? (That always helps, but may open his/her computer to attacks that could be more difficult to perform behind NAT -- so be sure you know the risks. They're small, but they're there.)
If you can find a port where data can be exchanged, you might be able to force WebRTC to talk on that port. OTOH, I can't remember any well-known UDP port apart from DNS's 53. Your best friend will be Wireshark and logging of SDP data from WebRTC.
If you can find a way to exchange UDP data with your friend, you might be able to trick WebRTC into using that. Perhaps even file a better bug report than 'it doesn't work, help', which is basically what Chrome's devs can see in your post. How much is your university network locked down? How? Does Virgin block any traffic? Those are some things someone more knowledgeable than me might ask.
Put yourself in shoes of an expert: Could you help yourself with the data you wrote here, if you could access neither university's nor friend's network to figure out what goes through and what does not?
Regards,
Ivan Vučica
via phone