Suggested Keychain password not saving on iPhone

Hello! I'm trying to change the password of one of my accounts (that Apple Passwords has advised me to change because of a data breach).

I go to the change password page of the 3rd party website, Apple auto fills a suggested new "strong" password (in both the "new password" as well as the "confirm password" boxes. I hit the change password confirmation box, but then....

Keychain Passwords do NOT save the new password. The little "update password" prompt does NOT come up. This means I've changed the password to some crazy secure complex password, but it has not been saved by icloud. Happens over and over.

What is one to do? Have you experienced this?