As far as AmEng goes, is there any difference in using either homework, schoolwork, or assignment to call schoolwork given to students to be done at home? Can these be used just about interchangeably?
ASSIGNMENT vs. HOMEWORK
An assignment is a task that someone is given to do, usually as part of their job.
My first major assignment as a reporter was to cover a large-scale riot.
An assignment is also a piece of academic work given to students.
The course has heavy reading assignments.
When class begins, he gives us an assignment and we have seven minutes to work at it.
In American English, an assignment is also a piece of work given to students to do at home. (emphasis is mine.)
Work given to schoolchildren to do at home is also called homework.
He never did any homework.
HOUSEWORK vs. HOMEWORK
Homework is work that school pupils (Chiefly BrEng)/students (Chiefly AmEng) are given to do at home.
Have you done your English homework?
Housework is work such as cleaning or washing that is done in a house.
She relied on him to do most of the housework.
: work that is done in classes in a school or given to students to do at home.
Point to all of this is, if some native speakers of AmEng actually use exclusively "assignment" (or schoolwork) for "homework" in the sense "schoolwork done at home," wouldn't the reason for that be that "homework" might have a quite different meaning in their vernacular, e.g. "housework" or "work done on the home"?
Homeworker in the US is a person who works on homes or a person who works in homes (maid). It is not a "clear" phrase at all because it tends to have ambiguous or no/multiple meanings. It is certainly something that I would expect to hear from a foreigner in the US. (emphasis is mine.) RyeBread - ELU
Please, consider the following example:
I'll help you with your homework/schoolwork/assignments once I'm done with the housework. (standard AmEng)
I'll help you with your schoolwork/assignments once I'm done with the homework. (nonstandard regional AmEng?)