Well, the purpose of troubleshooting is to definitively determine what the problem is and what caused it. Sure, it might have been caused by the iOS update, but that doesn't give you enough information that can then be forwarded to Apple for additional assistance.
All a username and password field is is a text box that you click a button and it sends another HTTP request similar to the one sent just to access the page in the first place with your credentials. That's it. If you can establish a connection to the website and get to the login page, that leaves an incredibly small subset of things it could be.
By trying a different network, we ruled out that your IP was blocked from authenticating to those sites. By trying different browsers, we ruled out that there was a faulty extension or some other bug with the browser. This leaves the input that you are supplying to the application, save for some bizarre scenario where your phone is infected with some extremely sophisticated targeted malware and modifying certain HTTP requests or a critical iOS bug, but to troubleshoot that further you'd need to MITM your phone's connection with a tool such as Burp Suite.
Speaking of MITM, this is incredibly unlikely but we're apparently getting into that territory now, are you sure that you're navigating to the correct website? Check the domain and ensure that you aren't entering credentials into a fraudulent website (which often pretend to reject your credentials to delay suspicion). Check your trusted root certification authorities in your settings.