You cannot see other people’s Pokemon nicknames at gyms. The only reason you can see the nicknames in pictures is because it is their pokemon. As stated in the link below, the pokemon nickname can only be seen by you.
Do Pokemon nicknames show up online?
No long answer for this, really. They can see your nicknames.
Is it good to nickname your Pokemon?
It’s one thing to name your new pal something with personal meaning; it’s much less special when your friend trades you a shiny Bulbasaur with the name “Fred.” Still, Masuda makes a strong point. “Please, give them nicknames,” he says. “You wouldn’t call your cat or dog ‘Cat,’ ‘Dog.
Can you get rid of Pokemon nicknames?
You can’t delete a nickname, so don’t give ’em one if you don’t want one! In Black and White, you can go to the name rater, tap the sprite next to the name of your pokemon and it will change it to it’s original species name, then when you evolve it, the name will change acording to the pokemon’s new species.