As my page indicates, I am an attorney. Mostly maritime law, but immigration as well.
There are exceptions, of course, as to whether or not the children acquire US citizenship at birth. The mother's absence from the US will pertain here. I don't think there's any doubt that the children may opt for US Citizenship at age 18, if they can make it to the States.
The main issues here are that 1) - Saudi doesn't recognize dual citizenship to begin with, and 2) - they don't recognize any rights except Saudi male rights. It'll be almost impossible to get the kids out of the country - they can't just walk up to a airline counter and hop on a plane there.