Check your internet connection first and switch between WiFi or mobile data to test stability.
Sometimes the app store or Google Play server causes the hang, so try clearing your cache or reinstalling.
Force closing and relaunching the app often triggers the update properly if it stalls.
Make sure you have enough storage space; updates often fail silently if the device is full.
On iOS you may need to restart the App Store app or even restart the phone to kick the download.
On Android clearing both the Play Store and Play Services cache helps in many cases.
If none of that works, check Supercell’s server status via their Twitter or forums.
Often an update may be region-locked and require a VPN to download temporarily.