Is it permissible to use zakat funds to establish a mortuary washroom affiliated with a mosque, purchase an ambulance, and cover its operating costs, such as wages and fuel?

Zakat has specific categories outlined by Allah in His Book, and establishing a washing facility for the deceased is not one of them. Therefore, it is not permissible to spend Zakat funds on it.

This washing facility can be established through general charitable donations (sadaqat 'ammah). The expenses for preparing the deceased are to be covered from their estate, and these expenses take precedence over the rights of the heirs. If the deceased has no wealth, then the expenses fall upon those who were obligated to support them. This is based on the statement of Sheikh Ahmad al-Dardir in Al-Sharh al-Kabir: "And it, meaning the shroud and accompanying burial expenses, is obligatory upon the one who was supporting the deceased due to kinship, such as a father or son." And the statement of Al-Nawawi in Al-Majmu': "If the deceased has no wealth and no spouse, then their shrouding and all other preparation expenses are obligatory upon whoever was obligated to support them, whether a parent or a child."