Is a wife forbidden to her husband if she says, "I don't want to disclose my son's illness," intending to conceal the secret of her husband's departure, and what is the way out of this situation if such a prohibition has occurred?

Abstaining from swearing by other than Allah is in compliance with the Prophet's (peace be upon him) saying: "Whoever has to swear, let him swear by Allah or remain silent." The statement "then I am ready to declare my wife unlawful" is a promise of declaring unlawfulness, which has no consequence unless the husband carries out his promise. Consequently, your wife is not unlawful to you, and you must repent and beware of repeating such an action.