I know nothing about the green card route except that without one you won't get your foot in the door.
Normally to get a FAA ATP you have to hold a FAA commercial and to get the commercial you have to have a FAA private. Holding a foreign ATP allows you to short cut that process. You'll have to have your CAA verify your licence to the FAA to jump the line. Other than that, it's the same as what an American pilot goes through. You'll have to complete an ATP Certificate Training Program which is 30 classroom hours and 10 hours in a sim, pass the written, get whatever training it takes to get ready for the checkride (if it's not a type rating you can do the checkride cold but I wouldn't suggest it) and pass the checkride.