The fact that the two responses so far bracket the options from very simple, to very thorough and realistic, with my plans in the middle, makes me pretty comfortable.
Details are, the arm was shattered in a killing crit - the rest of the description was "bone severs arteries, character dies in 9 rounds." When there's a (rare) killing crit in my game, I weigh a lot of factors quickly before deciding whether to modify, or let it stick. In this case, I modified. The game session ended right after the fight, so I've had time to plan the exact consequences.
I'm giving the character infrequent periodic RRs (6 hours) vs disease, and if they fail their penalty on actions increases due to "whatever/everything." - that's where I'm not worrying about neuropathy vs. inflammation vs. sheer pain vs. (other medical stuff I'm not an expert on, or that would distract from play.) When they hit a certain point, the arm becomes a life-threatening issue. If the cleric thinks to do Disease Purifications periodically, then these RRs aren't necessary. The character just stays at their starting penalty, and the arm atrophies and dies in a few days.
Big-picture, this now IS the adventure: save the barbarian. The party has options (but not herbs, and the cleric's blind from a different crit in the same fight) and a race against time. They can send a member for help, from a village the party thinks hates them (but doesn't, actually). The best possible outcome is the barbarian loses her arm, either the arm died or the party cut it off to save them. That opens the door for a search for a lay healer who can make a prosthetic arm, and a new twist on the barbarian's character.