Why does scanning for carriers take so long? Whenever I want to switch my carrier (even to back automatic selection), my mobile (rooted Galaxy S4 mini if that matters) scans one or more minutes for carriers. Why does it take so long? Wi-Fi Matic doesn't seem to require that much time to identify cell towers which in turn provide information on the available carriers, so what's the problem, and can this be accelerated somehow?
Your cellphone searches for a predefined (by the vendor) time (in your case about 1 minute) for available carriers and then displays the result. The delay is there to make sure there are as well the towers available for you to choose from which have a weak signal and loose some packages on the way.

You can not change that behavior in your Galaxy S4 mini.