The thing about Nanami, at least in this fic, is that he understood Geto's AND Gojo's reasonings for defecting, though he didn't agree with their actions afterward. He doesn't like Jujutsu society but was pulled back into it after Gojo's betrayal - partially out of obligation to protect, because he's a grade one sorcerer, and also the replacement Higher Ups were likely concerned that Nanami would join them since he was familiar with Gojo and Geto and had already left. Their lives on the other side were very difficult. Life in Japan still went on, but it was markedly different and not normal, so yeah, Utahime and Nanami really bonded.

I would also say he was incredibly protective of Utahime (along with Shoko) because of how she was treated by the Higher Ups and high-ranking sorcerers/clan members. He would've been the first to step in when she and Naoya got into it. Not to mention I envisioned them being the two teachers closest to their "students", though the schooling wasn't standard. They were mainly focused on survival, especially since Gojo and Geto did a solid amount of damage to Jujustu society, hence why they're working toward "peace" with the clans and other sorcerers. They've already thinned the ranks a lot.

Nanami would be both an asset AND a threat to Gojo when it comes to Utahime because he would represent something very important to her that Gojo has done his best to help her forget. And Gojo better remember that HE was the reason she and Nanami built that bond. :))))