OGNina would've died either way apparently, the GDs involvement just gave them a reason to execute her sooner. Why did she need to die? Well... because OGNina is the saintess's substitute/replacement but so long as the current one is healthy and protected, the substitutes existence according to spoilers bring negative health effects to the saintess (forgot what they do specifically. But it happens upon contact and related to both their divine powers). Since the king was already in love with Seraphie on the first timeline, upon realizing the truth from the pope (closest we'll probably have as the big bad and is revealed to be Nina's older brother) *the negative results of having the saintess and their substitute stay together*, the pope unaware of Nina's identity (he notes this in the epilogue how he has never considered the fact she is his sister in the epilogues) and also being in love with Seraphie chose the latter and since OGNina conveniently did a crime and was a perfect scapegoat, she was executed.
Additional spoilers (this was all according to the scholar Beato's POV. It is questionable whether or not he remembers the events of the first timeline)
For the 3 men in Nina's life
Dio, the red head the scientist/life saving doctor when asked by the king whether it is ok to execute a child, Beato claims he recommended Nina's execution (honestly, it was more the Pope's idea and Dio just validated it being a scientist whose mindset is to save the many)
The King ALLOWED and ORDERED Nina's execution
And Leo, was the one who executed her.
Seraphie/saintess was unaware of what happened after Nina was imprisoned. Beato says in her case, her sin was pure ignorance. Nina willingly helped her with the GD but apparently she conveniently forgot about Nina after Nina got imprisones and following the OGNovel plot ending, the kingdom just gets a happy end at the cost of the life of one innocent child who was eventually forgotten down to her name because no one cared about her (well, Beato to an extent got recurring heart attacks and nightmares after Nina's execution. So apparently, he and the pope *upon finding out in both timelines fell into despair* are the only two people who remembers Nina