I'll answer my own question, just in case anyone else stumbles upon this issue and wants to solve it.

With the help of Firebug, I saw that the noVNC application was telling my browser to do a 'GET /websockify', with that hint in mind I tweaked my nginx configuration and it now looks something like this:

upstream nova-consoleproxy {
    server 127.0.0.1:6080 fail_timeout=0;
}     

location /websockify {
          proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
          proxy_set_header X-Real-IP $remote_addr;
          proxy_set_header X-Forwarded-Host $host;
          proxy_set_header X-Forwarded-Server $host;
          proxy_set_header x-forwarded-proto https;
          proxy_set_header Host $host;
          proxy_http_version 1.1;
          proxy_read_timeout 86400;

          proxy_pass http://nova-consoleproxy;

          proxy_set_header Upgrade $http_upgrade;
          proxy_set_header Connection "upgrade";
        }

    location /console {
      rewrite ^/console/(.*) /$1 break;
      proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
      proxy_set_header X-Real-IP $remote_addr;
      proxy_set_header X-Forwarded-Host $host;
      proxy_set_header X-Forwarded-Server $host;
      proxy_set_header x-forwarded-proto https;
      proxy_set_header Host $host;
      proxy_http_version 1.1;
      proxy_read_timeout 86400;

      proxy_pass http://nova-consoleproxy;

      proxy_set_header Upgrade $http_upgrade;
      proxy_set_header Connection "upgrade";
    }