Steps to reproduce:
You will be autologged in on enwiki, but not on testwiki. You will be autologged in there as well once you visit the login page (maybe because that starts a new session). Happens with other wikis as well.
Might be related to the AuthManager deployment, but I wouldn't swear it; I have seen the not-autologged-in-until-you-visit-the-login-page behavior in the past.
Can't reproduce, I was auto-logged-in to test.wikipedia.org fine during step 2.
If you visited testwiki previously, you might have the local storage token set that indicates it already checked and shouldn't check again for a day, which exists to stop every visitor from checking status on every pageview. Special:UserLogin always does the check despite this flag. See anon-set.js and ext.centralauth.centralautologin.js for implementations.
If you can reproduce this, please capture the HTTP requests and responses to Special:CentralAutoLogin during step 2, which will both verify it's even trying and will let me start trying to figure out what's going on.