In what year (CE) did saying selichot in Elul begin? In what year were they made a standard addition to prayer books? Looking for an academic answer, not the yeshivahish one. And did Sephardim start doing it before or after Ashkenazim?
Rabbi Dr. Immanual Shochat's introduction to his translation of Slichos (page xii) places the first text of Slichos to Rav Amram Gaon. Wikipedia contends that the slichos part is of later vintage, but brings no evidence for the assertion (that the entire thing is whole cloth of later vintage, rather than interpolation of later slichos into what Rav Amram Gaon originally had), and links to one source which may or may not say that (it isn't clear how much of that paragraph the source is intended to cover).
Rav Hai Gaon is quoted in the Tur (O.C. 581) as mentioning the practice, and Rabbi Yitzchok ibn Ghiyyat is quoted (by the Rosh and Rabbinu Yeruchum as brought in the Bach there) as the first mention of specifically saying Slichos in Elul - earlier references are to between Rosh Hashana and Yom Kippur.
So that would place them at or after the 9th century (Rav Amram) or 10-11th century (Rav Hai). If you go with ibn Ghiyyat (if you were looking specifically for Elul), then that would be 11th century. He is also the author of many piyyutim.