The Adversary must wait until their own turn to use the Disappear Action.

This is supported by the following:

• On page 23, under "Disappear," it states:

  "If you are Revealed, you may move your figure from the main board to the same space on the mini-map, updating your Shadow token to where you Disappeared (even if the space is Bright). After Disappearing, you may not Ambush or use your Attack card the rest of the round."

• The Disappear Action is listed as a "Core Action" available to the Adversary during their turn, and actions are only performed on the Adversary's turn:

  "The Horror always has these Actions available: Move, Disappear, and Break Door. Each may be performed during the Adversary's turn once."

• On page 28, under "Adversary Turn Actions," it clarifies:

  "After all the Investigators have taken their turn, you may take yours."

There is no rule in the provided context that allows the Adversary to take the Disappear Action outside of their own turn. Therefore, if the Adversary is Revealed by entering or being on a Bright space during an Investigator's turn, they must wait until their next turn to use the Disappear Action.

Source: pages 23, 28.

Source: Core Rules (pp. 12, 22, 23, 28, 8)