The issue manifested for a subset of users as soon as they rebooted after the upgrade, with the passcode screen taking up to several seconds to register taps. The problems can persist once you're into your iPhone or iPad whenever using the keyboard. Key taps don't register for several seconds and then appear in a rush, making it very difficult to type and extremely frustrating.
If you aren't experiencing this problem, there's no need to go through these steps.
To fix this, all you have to do is reset your keyboard dictionary. I don't know why this works, but several people on Apple's support forums report that it does, and our own (and MacCast's own) Adam Christianson confirmed it. Here's how you do it:
Open the Settings app, go to General, and then Reset, which is at the bottom of the General screen.
Settings > General > Reset
Then go to Reset Keyboard Dictionary, which is also at the bottom of the screen. Tap that and you'll be asked to enter your passcode. Note that this will delete any and all custom words you have entered on your keyboard.
Reset Keyboard Dictionary > Enter Passcode
That should do you!