I'll defer to those who are more familiar with curriculum history in the United Kingdom, but can offer that students were studying film in American schools as early as the 1930's. A number of New York City teachers collaborated with local filmmakers to organize "film appreciation" courses and events.

In one instance, a teacher at Benjamin Franklin HS worked with filmmakers to secure underwriting from the Rockefeller Foundation to show general release films, as well as educational films as part of a humanities course. Although the course didn't look like a modern film studies class, students watched films such as The Devil Is a Sissy (1936), Fury (1936), or A Passion for Life (1951) with the intent of talking and writing about the relationship between the films and current events. The all-boy student body represented dozens of different cultural and ethnic groups and school leadership felt share experiences around popular films would provide a common and shared experience. The funding allowed the school to show the films in their auditorium, similar to how the public would see them. In effect, this served as a workaround for any copyright issues as the movies were considered public screenings.

That particular program ended in the 1960's but by then, "film studies" classes appeared at high schools throughout NYC and the rest of the country as an elective part of the liberal arts education. There are multiple references to "film studies" appeared in articles about high school curriculum and on the agenda of national English teachers conferences throughout the 60's and 70's. By that point, the "fair use" exemption to copyright law was commonly accepted. It would take a few more decades for the cost of VHS recordings and VCRs to come down and by the 1980's, "film studies" was a common elective or part of the English curriculum.