Fixed. Wasn't a problem with Wedge, but with a quick hack I wrote for Lestrade's and backported to Wedge.org (the website) by mistake.
The original problem was IE11 was unable to load compressed CSS/JS files on the new server I had. I managed to make it work in Lestrade's, but the actual solution was to remove gz extensions from the 'static' file list in the nginx configuration (it's a reverse-proxy server). Nginx used to catch gz files and directly serve them with their original MIME type, when it was supposed to pass them to Apache, which through the htaccess file was setting the MIME type to the correct one (i.e. text files).
I had the nginx config file fixed, so the fix was no longer needed. But I forgot to remove it, so IE11 kept serving these files uncompressed. And because of a final mixup between two fixes, I ended up serving uncompressed extensions with compressed content, so it broke the site when I updated it last night.
BTW, IE still sucks hard. :P
I'm not even adding support for Edge... After all Microsoft themselves are marking it as Chrome-compatible. So, I'll just let them fix everything by themselves, to me, Edge is an older Chrome build and that's how I'm treating it. (And probably how they want others to treat it, because let's face it, anything is better than being treated as a "newer IE".)
PS: I just found out that my first IE11 fix for this was from late 2014, so it couldn't be linked to the new server. Uh. I'm at a loss here. I'll just revert that revision (c3c77a3b22223eedce97f18b7ad719437b3dc154), and hopefully it works everywhere.