Dear tubedogg & Gabriel,
As Gabriel pointed out, this problem is NOT related to iCloud Keychain, but actually to Two-Factor Authentication. Unfortunately, this problem has not been solved even by iOS 10.0.2 released on Sep. 23rd yet. If you automatically turned on the authentication during your update to iOS 10x, here is the solution.
STEP 1: When you update the iOS, skip "approve the phone" by pushing the button of Cancel. You can retry it later.
STEP 2: Complete the iOS update by following the instruction.
STEP 3: Make sure that your Two-Factor Authentication is ON at this moment by either of the following options: Option A) Read a notification email from Apple. Option B) On your iPhone, go to [Settings] --> [iCloud] --> tap your profile of Apple ID --> Password & Security --> Two-Factor Authentication
STEP 4: Turn OFF the Two-Factor Authentication. During this process, you will be required to reset your Apple ID (iTunes & iCloud) password. But no worries, no date will be deleted. If you received the notification email about the Authentication, you will find a link of "turning off the Authentication" in the email.
STEP 5: After completing the Authentication OFF, retry "approve the phone" by going to [Settings] --> the red flag. Without doing anything, your iPhone will be automatically and immediately approved. All set.
You can get the Two-Factor Authentication back ON anytime later if you wish.
Hope this helps.