You would normally at least be entitled to due process. My understanding is that Trump is citing a law that allows the President to deport nationals of a country we are at war with without due process, except we aren’t at war with Turkey.

I believe you are mixing up different cases. That is not the statute used at all for the Turkey case. The Alien Enemies Act you are referring to was exclusively used for the designated foreign terrorist organizations (i.e. TdA & MS13; Venezuelans who went to El Salvador prison).