You did simply change the Apple ID, correct? You did not create a new one? Did you follow these instructions?
Change your Apple ID - Apple Support
Try signing out of the ID in the settings, close the App Store and force restart.
Settings>iTunes & App Store>Sign out
Close the App Store. Press the Home button two times quickly. You'll see small previews of your recently used apps. Swipe left to find the App Store and swipe up on the app's preview to close it.
Next, force resort the phone. No data is affected by this. Hold down on the sleep and home buttons at the same time for about 10 seconds until the Apple logo appears on the screen.
Go back to Settings>iTunes & App Store>Sign In. Try an update again.