Nurse becomes the role of the first Doctor to die.
And since 2 roles kill, one prevents scum from making a nightkill, I wouldn't call it "protect", and should be better used on scumreads, unless we come up with a pattern that everyone agrees to.
I'm still thinking about it - going with a circle targeting method would mean that there's 2x 1/5 chance of killing the scum at night, so 36% chance to win the game right away, but 64% chance to kill two townies and since we're going in a circle noone is protected from doctor kills. That is if both killing Doctors are still alive, 4/6 chance (one death out of the 4 nonkilling docs, since nurse replaced the first death). 2/6 of the cases (one killing doc dead) we'd have 20% chance to win the game, 80% chance of killing a townie.
I don't think town is in a position that we need to take these chances of letting two towies die in exchange for a chance to win possibly faster, when we can make a plenty of lynches and pick proper night targets that are not blind guesses like this method.
One other thing I thought of - we'd unofficially vote one person (NL today), and he'd get targeted by everyone. Self target isn't possible probably, but that's not a problem, since if the jailkeeper is on the target he's safe anyway, if jail's dead or is the target than [norm cpr quack] any 2 of these being used means he's safe. So regardless of which doc we're missing (and possibly targeting) target is sure to stay alive, but we'll confirm him as scum or town, since no NK happens if the jailkeeper is still alive and he's scum, and even if it's the jailkeeper that we're missing, we'll still know if he's scum, since the weak doctor dies in addition to the NK. A scumcheck for free (target won't die if he's town) and scum can't deny the conf town because of the protections (unless Jailkeep is dead, or is the target). The only downside is that IF the jailkeeper is dead (1/6 chance) AND the weak doctor is the NK target (1/5 chance), than it'll look like the target is town even if he's scum. There's a 3,33% chance of this happening.
Well maybe I forgot to take something into account and this is completely useless, let me know what you think anyway.