After countless attempts I managed to get the builds running. You should first try to package your project in the Unreal editor in order to check out any error messages.
In my case it was a missing entry of a developer team in the Xcode project. Also make sure that you enter the ‘Apple Dev Account Team ID’ under ‘Xcode Projects’ as well as the correct ‘Bunde ID Prefix’ in the project settings.
What I didn’t test is if you need an explicit name for the app identfier or if it works with a wildcard name.