Henry Weber edition of Arabian Nights published in 1812 contains multiple uses of adieu.
Henry Weber, comp., Tales of the East: The new Arabian nights; Persian tales; Persian tales of Inatulla; Oriental tales; Nourjahad; and additional tales from the Arabian nights (Edinburgh: James Ballantyne, 1812), 7–8, 18, 76
She then parted with him. Adieu, covetous Simoustapha!
. . .
They sealed their adieus with tender embraces and new tears. In the name of Mahomet! said the impatient Namouna, have done, and let us depart quickly. The lovers parted.
. . .
I must remember that I am the son of the king of the Indies, dearly beloved by my parents, and accountable to their subjects for the presumptive heir of the crown. Adieu ! dear Simoustapha, said the queen, embracing him; adieu! prince, model of sovereigns, light of the world, and genius of every virtue!
. . .
You see I speak frankly. Adieu: I am called elsewhere; but I shall be with you early to-morrow. So saying, Cassanak bowed, and went away.