Yes, it's possible to infect (and infest) the nurse before the teacher, but it's unlikely. You have to go out of your way to get the catfight event, which isn't available every day, so you can visit the nurse. And on the day you do the catfight you obviously won't be able to visit the Sanctum, since they are mutually-exclusive daily school options.
The nurse is usually the last NPC that gets infested in my games. And skipping her has zero effect on your progress. IMO she's just not worth the effort. Unless you're a completionist who just has to see every image in the game.
As for infesting the teacher, you just have to wait until her resistance to the idea of having sex with you gets low enough. You have to go to the teacher's lounge to interact with her. If she's not there, and if her resistance has not been sufficiently lowered then you won't get the option to have sex with her. And if her resistance is really low (and her arousal high enough) she'll jump on you the instant you enter the lounge.