Do any other nations have an equivalent of the "Air Force One" call sign?
It is all in your link (in the "See Also"), E.g.:


Brazil: https://en.wikipedia.org/wiki/Brazilian_Air_Force_One
France: https://en.wikipedia.org/wiki/Cotam_001
India: https://en.wikipedia.org/wiki/Air_India_One
Japan: https://en.wikipedia.org/wiki/Japanese_Air_Force_One
Vatican: "Shepherd One", or just in Italy "Volo Papale #" (https://en.wikipedia.org/wiki/Air_transports_of_heads_of_state_and_government)


Note: Often it is better to disguise who the airplane is transporting (especially when flying above a lot of different countries), and often the president has not so much power ("nuclear football"), so a hired jet it is enough, no need to communicate home.