Q: Need help changing Apple ID email address

Try a forced restart of the phone and after it starts up, try again. Hold down on the sleep and home buttons at the same time for about 10 seconds until the Apple logo appears on the screen.

Exactly where are you trying to update the ID? Can you log onto the internet, get your email?

Maybe this helps: Change your Apple ID - Apple Support

If you are doing this on your laptop, have you tried closing the browser and restarting your computer? You might also try power cycling your router by unplugging it from power for about 30 seconds. Even if you are connected to the Internet, power cycling can or may help with problematic web sites.

I don't believe that it is a connectivity problem per se either, just a problem with that web site. Did you close the browser and reboot the computer? Have you tried clearing the cache in the browser, close the browser and then reboot the computer?

That's great. I'm glad that you finally got it resolved.