there is more ways to fix Install Button Unreal Engine is greyed out :
1 - goes to the bottom left and sign out of your account then go to the unreal engine libraly page now it's shown and select your version Press “Set as Current” back to login .
2 - Sign Out. Then choose Sign in Later .
3 - Press on any Tab then back to Unreal Engine Tap then choose Library .
4 - Press on the home button then press on the unreal engine button .
5 - Press on ue5 and Press on get early access and go back to library .
6 - enabled 2-factor authentication .