iOS 12.5.7 is the final software update available for older 32-bit and early 64-bit Apple devices. Because the modern YouTube app requires iOS 13 or newer, you cannot simply download the current version from the App Store. Attempting to do so results in an "Update required" error.
Note: If you have never "purchased" YouTube with your Apple ID, use a newer iPhone or iPad to download it once. This will add the app to your account, allowing the older device to access the "Purchased" download option. YouTube for iOS 12.5.7 Download - Apple Support Community youtube app for ios 12.5 7 download
"YouTube for iOS 12.5.7 download – App Store last compatible version" iOS 12