Here are some options for how to structure this and for where to put the focus:
Chronological Narrate the process as it happens. The important thing to remember is that while the loss of the arm is a foregone conclusion to you, to the character it is likely something that they never thought would happen. Losing a limb is like some strange fantasy becoming real. You can slowly build up the anxiety and the panic as their options fall away one by one, and then mix in the effects of fever and gangrene on their mind. At some point, while they are still relatively lucid, they will have to choose between amputation and death.
One way to end the episode is to make them wait too long. They miss the window where they are still lucid enough to get the arm amputated and they slip into a feverish state of half-conciousness. Their fear of amputation now twists into a fear of death, and they regret waiting so long. Then you can save them. Some stranger, maybe the doctor that refused them earlier, found them and performed the amputation. The character wakes up with the missing arm, but feels reborn rather than incomplete, because they had already accepted that they would die.
If you get this right, it will be a powerful, but challenging read. If you have a story to get on with, this may get in the way. The following are some lighter approaches.
Flashback Start the story with the arm already gone. Slowly reveal the backstory bit by bit, once the character is established. Show how the loss of the arm explains the character that we already know. Maybe they haven't quite resolved the trauma, and a catharsis awaits. Maybe they hold someone responsible for the loss and are looking for revenge as a resolution to their anguish (ref. Cap'n Ahab). Maybe going through the loss of a limb made them confront some of their earlier character flaws, like arrogance (ref. Jaime Lannister).
Jump-cut Foreshadow the loss of the arm and then jump forward to when it's done. Maybe you can set up the character's certain death just before the cut, to leave readers with a sense of mystery to resolve. Perhaps it turns out that they saved their own life at the last minute by making a deal with the devil, and they are now reluctantly on a quest to repay their debt (all the while trying to adjust to life sans arm).
Finally, you should probably be conscious of the fact that this is a serious injury that many people deal with in the real world. As such, getting this wrong, either by poorly characterizing the experience, or overly sensationalizing it, can come off as insensitive. The best way to guard against this is to do some research. Read about people who have lost limbs in combat situations (plenty of real stories there, unfortunately) and what the experience was like for them.