So this has been terribly frustrating. I have what I believe is the exact same problem, but I am going to quickly summarize.
I was up and running with iPhone 6, iPad mini, and iMac all updated with most current software versions (iOS 8.3 and OS X). Everything operating very smoothly. My university IT department requires us to change password every 90 days. My university uses google for everything, but to make sure everyone understands the process (because in reality I do not), I login to gmail with (username)@(university).edu and I am immediately redirected to another site to complete the login process (ours is called gozer.(university).edu). It does not even matter what password you put in on the initial google page, the verification only occurs on the subsequent page.
I do use keychain for all my i-devices and have never had a problem. When I changed my password this time, the 1st time with iOS 8.3, every google connected app stopped working (primarily gmail and calendar - I was able to google drive back up and running on iOS). The symptoms are similar to what has been discussed here, but they differ on how you try to login. When initially attempting to open mail and connect to my gmail university account, I get prompted that I must reenter my password, but their is no way to enter the password.
I then reluctantly deleted my university gmail accounts and tried to reset them back up. When I do this I can occasionally get to a login page, but the process hangs up and never completes. I have tried powering off several times and for extended periods. This occasionally does get my further into the login process, but the process always hangs up.
After reading this entire post, I went into every i-device and deleted every password on each device and my keychain that I thought was at all related to any google account (I have university ones and personal ones) as well as any password that had (university).edu in it. This process still does not work and I get the same error. I even tried moving the slider to off for "names and passwords" and "passwords & autofill" in safari settings.
As a last ditch effort I have tried configuring gmail manually as chopper26 suggests above suggests. This process does work and is a viable solution for gmail only. Unfortunately it does not address contacts or calendar so it is not a complete solution.
Any help would be greatly appreciated and yes I know I can use other apps (gmail app and google calendar app) to fill the void, but I am pretty stuck in my ways and wanted to use the normal Apple mail and calendar app. Thanks in advance.