From studying Tolkien's life, a British student could encounter Anglo-Saxon at school before university. I do not know if A-S was a mandatory part of the curriculum, but between 1904-08, Tolkien encountered medieval literature and languages while attending King Edward's School (grammar school, he was at the age of 12-16), and he was spurred on by a teacher to read and learn the original languages--Old English, Middle English, and Old Norse.

I hope that helps!