App installation is handled by iOS and by the App Store that is run by Apple. In essence, Apple handles the purchasing, provisioning, and installation of all apps on your device. If you are having issues with the app installing, make sure you are on a good Internet connection and if the problem persists, restart your iOS device.
If the app does not appear on the home screen, you may also want to restart or not appearing on your home screen, we recommend restarting your iOS device. If the problem persists, please contact the Apple App Store Support for assistance.
In the interim, you can search for OnSong by swiping down while viewing your device's home screen and search for OnSong. If OnSong does not appear, it may not be installed. Otherwise, you should be able to open OnSong from the iOS search screen.