He does the Black Sacrament, but nothing happens until he returns to Hogwarts, where he is visited by Sally Anne Perks.

Turns out she is a dark brotherhood assassin. When the killing curse rebounded and hit Voldemort, he should have died but did not. This attracted the attention of the Night Mother, who sent Sally Anne Perks from Tamriel to keep an eye on Harry and this new world, by disguising herself as a Hogwarts student. After hearing that Voldemort has returned, she decided to give the book to Harry. But as Harry asks her to kill Voldemort, she is unable to due to his horcruxes. The two embark on a journey to destroy them instead.

However, Harry can still name one person for her to kill. For a price.

This by the way also attracts The Order of the Black Worm to this world, and they take a special interest in Voldemort and his inferi.