If the Lunatic Devotees are killed and Cultist Archers are lured away and killed far away, the Lunatic Cultist will not spawn. The Cultists will not respawn, either. This can be fixed by reloading the world.

What happens if the lunatic cultist doesn't spawn?

The cultists won't spawn if you placed walls in the dungeon's entrance. If you did, make sure you destroy them using a hammer. If there's a Meteorite nearby, Meteor Heads will prevent the Cultists from spawning.

How do you get lunatic cultist to Respawn?

The Lunatic Cultist locks on to every player, so if one player uses a Magic Mirror or an Ice Mirror it will despawn. If the player loses against the Lunatic Cultist, the Lunatic Devotes and Cultist Archers at the Dungeon will respawn after one in-game day passes.