The council is alerted of the slayer's death approximately sixteen hours after it occurs. As always, they gather their forces and observe the proper rites and rituals. They determine that the new slayer is located in Sunnydale, California, above a suspected Hellmouth. Rupert Giles is selected to serve as her watcher, due primarily to his expertise in related mystical environmental phenomena.
Strings are pulled. Within two days he is in Sunnydale. Within two weeks, he has been hired as the new librarian for the local high school, which the new slayer - a girl named Julia, he's been told - is registered to begin attending in the fall. He does not make immediate contact with his slayer. She's not been isolated from her family, and while she'll be having nightmares, it's very likely that she hasn't yet noted the onset of other mystical powers.
He waits for his charge to come to the library in search of this year's textbooks.