Ok, so it seems like there have been many changes to the Authentication framework of MediaWiki, which is why this extension stopped working. I'm currently trying to update it, but so far I have no idea what I'm doing. I did set some excellent pointers from @anomie in the IRC, which I've pasted below. Perhaps they might prove to be more useful to someone who does know what they're doing.
I see right now it wants $wgAuth set to an instance of the AuthWP class, and it also hooks UserLoadFromSession and UserLogout. The UserLogout hook is fine, you can keep that without issue.
The UserLoadFromSession hook is deprecated, if you want to keep that functionality you'd need to create a SessionProvider to do it. The tricky part there is that right now a SessionProvider has to either accept arbitrary logins (which you likely don't want) or cannot support any login or logout while it's in effect (which would also be problematic). I should probably make a way to have a SessionProvider that doesn't allow arbitrary login but still allows logging out.
As for the AuthWP class, that looks like it should be a failry straightforward subclass of AbstractPasswordPrimaryAuthenticationProvider. https://www.mediawiki.org/wiki/Manual:SessionManager_and_AuthManager/Updating_tips#The_AuthPlugin_and_AuthPluginUser_classes describes what the AuthManager equivalents are for each of the AuthPlugin methods.
I note that allowing logging in to both systems via either system's interface is going to allow bypassing of security provisions in one or the other's login process. For example, if MediaWiki is configured with a two-factor extension, someone could log in to WordPress without two-factor and be logged in on MediaWiki too. (that's regarding the UserLoadFromSession hook function)
Honoring the WordPress cookies may also break AuthManager's flow if MediaWiki has any SecondaryAuthenticationProviders that need UI, since I expect wp_signon() sets WordPress's cookies right then where MediaWiki wants to keep using its current session to finish the process.