I would really like to take a look at this but having just moved house the only internet I've got is at work, or my mobile. That and scant time, since it's something of a project house and I seem to be spending my spare time filling skips with the junk the last owners left!
If I remember rightly, the authentication "bridge" with the forum was done with this extension:
https://www.mediawiki.org/wiki/Extension:SMF/Users_Integration
(If someone takes a look at the forum's LocalSettings.php file, if it matches what's described on that page then that's it).
As a first step, has everyone tried clearing their cache and cookies on their browsers? Because the cookies may retain some identification against the old host which won't work on the new one.
The move sounds like hard work ... so long as there's a nice big shed at the end of it which you can fill with your own junk, I'm sure it will be worthwhile!
I tried clearing the cache and cookies, but it made no difference.
I managed to view the LocalSettings.php file and found the lines ...
# This requires a user be logged into the wiki to make changes.
$wgGroupPermissions['*']['edit'] = false; // MediaWiki Setting
Which appears at the top of the code you linked to, but after that it appears to my incompetent eye to be different.
Looking at things logically, if it worked before, it should work now. The only differences between the old and new server are what ever stopped it working in the first place and the changes Mark made. I think what Mark did, possibly amongst other things, was to add a redirect. So could the code linking the wiki and forum be looking in the wrong place for the wiki?