I've changed my main Apple ID email, but this hasn't updated in various parts of my iPhone. How can I fix this?

I started using my new iPhone with my old Apple ID, which was tied to a very old and cluttered email address (this was set up back when I got my first iPod Shuffle!), but soon decided that I had better change the email account. I changed the primary email for my Apple ID online, which seemed to work fine, and many services (e.g. iCloud online, Apple ID management) have made the switch seamlessly.

There are others, however - such as iCloud on my iPhone - that are stubbornly using the old email address, and not letting me change it use the new one to log in. Entering my new password (I changed that too) on such occasions sometimes unlocks the service, but sometimes doesn't (as was the case when trying to log into Facetime).

Has anyone had a similar experience, or would someone be able to suggest something to sort this out? I have tried turning the phone on and off again (just in case) and have logged out of all browsers (I wanted to log out everywhere, on everything, to see if that would fix it, but couldn't find a way), but this hasn't corrected the issue where it has arisen.

Any help/suggestions much appreciated 🙂