Normally I don't worry about natures, but I want to be extra cautious, because Emerald's a lot harder than Sapphire, and Tsunami(Mudkip) still nearly died like 5 times in my Sapphire Nuzlocke. I know there's a best nature for mudkip question, but that's for a normal playthrough(presumably, it's never actually said in what way this Mudkip is used), and I was wondering if it being a nuzlocke would change things. Thanks!
It all depends if you want your Swampert to hit harder or take hits better. Through my personal experience, a Relaxed nature (+Def, -Spe) would be better for taking hits better and either Quiet (+SpA, -Spe) or Brave (+Atk, -Spe) if you want an attacking one. Swampert already has low speed, so the slower speed won't make a difference most of the time. For the relaxed nature, most early game Pokemon have physical attacks, including all of Roxanne's Pokemon and all of Brawly's Pokemon. Also, the only attacks on Flannery's Pokemon that hit you for neutral damage at that point would be physical attacks. Brave and Quiet are pretty obvious. Don't go for Adamant unless you're sure that you won't be using Surf or Ice beam. An adamant nature may make it so that you miss out on crucial KOs when using your special attacks.
Overall, I'd say go for Relaxed as it may help in the early game quite a bit and while there are a few special attackers in the late game, you should be able to get a few good encounters to help against them.
Hope I helped.