Apple released iOS 8.3 with lots of bug fixes and enhancements however, a new bug have surfaced affecting quite a number of people using iPhone 5S to the iPhone 6S.
This bug was noticed by a lot of people when a request was made to authorize an app with the Touch ID before download begins the result was that Touch ID wasn't responsive to the authorization request.
One way to get this fixed is to go to Settings and click on Touch ID & Passcode. Enter your passcode when asked then switch off iTunes & App Store and then switch it on again.
If your already registered fingerprints were deleted after the update, you would need to re-register your fingerprints.
Once the above mention steps are completed, the issue should be fixed as reported by some.
Apple may eventually release a fix that will address the issue in the next iteration of iOS 8.3