Taunt makes the afflicted Pokémon unable to use status moves. The effect lasts two turns or until the subject is switched out. Unlike most status moves, Taunt will affect a target with a substitute. On the turn that Taunt is used, if the Pokémon using Taunt goes before the target and the target selected a status move that turn, the target's move will fail.
If both Pokemon use Taunt, the faster one's Taunt will work and the slower one's will fail.
Source: Experience and this
Hope this helps!
The faster one would block the slower one from using taunt. example turn Tapu Koko used taunt! Hawlucha fell for the taunt! Hawlucha can't use taunt after the taunt!
Source:Exp
Whichever goes first will Taunt the other Pokemon, thus preventing the second Pokemon from using Taunt. It's impossible for two Pokemon to use Taunt on each other in the same turn.
Source: Experience and Bulbapedia
Hope I helped!