The child can then register in school. You have to know the facts and circumstances of each individual case. It is very difficult to conjecture. It is possible to go to grammar schoool and high school in America while being an illegal alien. But it is virtually impossible to do so with reference to college.
Colleges have to register all foreign students.
There has been some controversy about this in the American press since some students have been illegal since early childhood and know no other country. Students can be here for years on student visas. Scholarships are available to gifted high school and college students. But you have to go through normal channels. I know since I have brought nearly thirty students from Ukraine to study here.