Update to iTunes 12.2 via App store
The app store depends on network responses and if the server is overloaded (as it is when big updates roll out), you can get caught waiting.

If you're not in a rush - set a timer and try again in 8 and then again in 24 hours.

If you are in a rush, force quit the App Store app and open terminal app.

You can run softwareupdate -ai and let that window wait as long as it takes for the server to respond. That is not going to hang your UI like the App Store app can. It won't be any faster, but at least you can do other things while you wait things out.