More recently than many people realize! In the current forms, they're really a product of the late 19th and early 20th century. At least in my own tradition, Lutheran, Sunday School was viewed with considerable suspicion - the fear was that it would lead to a decrease in religious education, not an increase. At the time, so many Lutheran churches had an attached school, and the students would receive religious education during the week along with all the other subjects, and some people feared that parents would see Sunday School as an excuse to neglect classes during the week, as well as their responsibility to teach their children themselves at home. Luther's Small Catechism specifically included headings that it was intended to help the head of the household to teach his family, not necessarily for pastors.

Bible study is considerably older, though. Of course, studying Scripture has been common and encouraged for most of Christian history, but I assume you're talking about today's "small group Bible study" set up where a small group gathers to read particular passages and discuss them, rather than study of Scripture in general. I can't think of any examples from the Early or Medieval Church (doesn't mean it doesn't exist, of course) but in the 17th century something akin to that arose in the Reformation. In Lutheran lands, the movement known as Pietism happened as a reaction against the sometimes-arid intellectual emphasis of Lutheran scholasticism, and one feature was increased pious Bible study, especially in small gatherings in homes. These gatherings were called "conventicles" and could reasonably be considered the ancestors of today's Bible studies. Conventicles also occurred in Britain as a reaction against the established Church of England, but I hesitate to get into that because it's more political in some ways and getting out of my field.