Question
Is it forbidden for observant Jewish students in the Diaspora to study on Saturdays?

Answer
I assume your question refers to secular studies since it is clearly desirable for Jews wherever they live to study Torah on Shabbat and on weekdays. It is permitted on Shabbat to read secular school books but ,if possible, this should be minimized (Shulchan Aruch Orach Chayyim 307 17; Mishna Berura 307, 65). With rare exceptions (Shu"t Chelkat Yaakov Yoreh Deah 190), it is not permitted to attend school- where a gentile teacher is writing on the blackboard and performing other labors- on Shabbat on a regular, planned basis (Mishna Berura 340, 21). Technically, it is almost impossible for a student to be present in such a situation while observing all the laws of Shabbat (Mishna Berura 340, 21).
All observant Jews, students and others, should be aware that continued living in the Diaspora at this time, when Aliyah to Israel is possible for most Jews, is halachically problematic.