The Shulchan Aruch speaks of women. Men, who have no actually obligation L’Halacha to go the mikvah may shower after going to the mikvah. R’ Wosner writes this explicitly in Shevet Levi.
There are many who Mattir taking a brief shower as long as the intention is not to get the mikvah water off but some other intention like the mikvah floor is dirty…
Please note that just because you couldnt find a Halacha in the shulchon Aruch doesnt mean it doesnt exist. Did you check the Nosei Keilim as well as other Halachic sources?
As stated above a person has no obligation to go to the Mikva (nowadays) so the Inyan doesn’t apply. Actually according to Kabbalah (Reb Chaim Vital), one should shower after the Mikva.
I believe this is an inyan related to women on their leil tevilah. And even in that case, there is a heter for certain women to shower with some type of chatzizah on them, like a bandaid.