Q: System Preferences hangs on adding Google account

I am running OS X Yosemite 10.10.3 – the problem I am about to describe started to occur just after installing the latest software update – on a MacBook Pro (Retina, 15-inch, Mid 2014). All of my applications are up to date; Mail is version 8.2 and System Preferences is version 14.0 if that can matter.

After installing the latest Yosemite update and restarting, I started to see a notification asking me to enter my Google account password in System Preferences. I clicked yes, and was brought to Internet Accounts and presented with a pop-up dialog looking like this:

The spinning animation never stopped – I waited at least five minutes. I also included Mail, in which this pop-up also appears when trying to configure certain email preferences.

This was fine, although the notification, which continued to reappear occasionally, was somewhat annoying. However, then I decided to remove my Google account from Internet Accounts and re-add it. I then realized, having removed the account, that I am shown the same non-functional pop-up dialog when I try to add a Google account. (Adding iCloud accounts appears to work fine.) Consequently, I cannot access any Google services on my Mac anymore. (Everything works fine when using mail.google.com, calendar.google.com, etc.)

As I mentioned previously, the dialog also appears when selecting Add Account in Mail.

Unfortunately, I had no backup within the last few minutes to which I could revert in order to restore my Google account.

I assume that the problem is caused by some change in Apple's protocol for interfacing with email in the last Yosemite update, as everything works exactly as expected on my second Macbook Pro, which is running all of the same software except that I have not installed the latest software update. (To be precise, this includes iTunes 12.1.2, OS X Update 10.10.3, and OS X Yosemite Recovery Update 1.0.)

What can I do?