I completed this run a bit over a week ago, and I hadn't been trying for a deathless run; it just happened. After thinking things over I realized that this run should be replicable in theory, barring really unfortunate critical hits. I have since deleted that save file, so I don't remember all of the natures of the team members. The only one I forget of the core members is Snivy's nature. Some of the team members could have had better natures than mine did anyways, but I will just list the natures mine had.
A guide to a deathless Nuzlocke does feel a bit like it betrays what a Nuzlocke run is, especially since there is some manipulation involved in getting the team earlygame. I will list the core members of the team, and then the other members who I had but didn't feel necessary. The leveling I used had the party members I was using at level 20 for Gym 2, and adding 5 levels for each gym. Elite 4 was hit at level 55.
Core Members
Snivy (Calm): This is one of the starters, and is therefore a gimme. He has a decent amount of bulk for taking hits. The set I ran on him was Leech Seed/Coil/Return/Leaf Blade by the time the game ended.
Lillipup (Lax): Lillipup comes early enough in the game that he could also be considered a gimme, if you don't have any qualms about resetting. Lillipup was able to take hits that I didn't want anybody else on the team to take, and was also my goto mon for most of the game for things like Mushnarna. Mine also didn't get Intimidate; I would imagine he would have been more helpful with it. It ran Crunch/Return/Dig/Aerial Ace by the time the game ended.
Drillbur (Hasty): Drillbur can be made to appear in Wellspring Cave by using repels and running into dirt clouds until you see one, which makes him a gimme. After he gets Dig he begins to sweep easily. I held off on evolving him for 2 levels so that he could learn Earthquake earlier. It ran X-Scissor/Earthquake/Rock Slide/Swords Dance by the time the game ended. It never needed to use Swords Dance, and X-Scissor was interchangeable with another TM if the situation arose (it never did).
Archen (Quirky): Archen is a fossil pokemon, and is therefore also a gimme. His high attack and great speed will allow him to have few problems with opponents. He can probably run special attacks moderately well if you want him to, but I preferred to use his much larger attack stat. By the end of the game he was running Acrobatics/Rock Slide/Crunch/Dig.
Underwhelming Team Members
These are the other members I had on my team. I found them underwhelming for different reasons, and believe that they could be replaced to allow for a more efficient run.
Sandile: This guy's problem is that the core members have enough overlap to make him redundant. There was never really any situation where I wanted to use him over one of the core members.
Vanillite: I honestly don't know how this pokemon made it onto my team. It has a horrible movepool, and is lackluster until it reaches Vanilluxe, at which point it is only average. It could definitely be replaced with something useful.
Other Team Members
These were team members that I used for a short period of time, and feel deserve to be commented on.
Patrat: This one I actually found useful in the earlygame. Its main purpose was to be evolved into Watchog and use Rock Smash against Lenora, and Detect to guard against a boosted Retaliate. He was dropped after that, so I don't know how he would have performed past that point. His Detect was really nice against the earlygame Patrats that use Bide.
Reshiram: This member was quite helpful in the Ghetsis fight. What I did was pump Reshiram full of X-Specials and X-Speeds and had him sweep Ghetsis. I would imagine that Zekrom would be able to do something similar.