Parents who wake up in the middle of the night in order to cover their children or give them a pacifier, in principle need not wash their hands because covering a child or putting a pacifier in his mouth does not require touching the child’s mouth or any other bodily orifice.
However, if one wakes up to prepare food for the child or change the child’s diaper, it is proper to wash one’s hands prior to that, in order not to touch food or one of the child’s bodily orifices with unwashed hands. Likewise, it is proper that a woman who wakes up in the middle of the night to nurse her child wash her hands before starting. However, if it is very difficult for her to go wash her hands, she may rely on the lenient opinions that do not obligate one who awakens in the middle of the night to wash his or her hands (Eshel Avraham Butshatsh 4:1; see also halachah 4 which mentions that some say that nowadays the ruach ra’ah does not exist). In any case, according to all opinions, no berachah is recited on washing performed in the middle of the night because the Chachamim instituted a berachah only on the morning washing, which prepares us for prayer and the new day.
L’chatchilah it is proper that a person who wakes up in the middle of the night to drink wash his hands three times before reciting Shehakol. Similarly, it is proper that one who wakes up in the middle of the night to relieve himself wash his hands three times so that he can touch his bodily orifices without concern. After relieving himself, he must wash his hands so that he may recite Asher Yatzar. If he wishes, he may refrain from washing his hands prior to relieving himself in the middle of the night, by relying on the opinion of those poskim who maintain that a person only needs to wash his hands three times after rising in the morning. However, after relieving himself, he must wash his hands in order to recite Asher Yatzar. If he does not have water, he must clean his hands, for example by rubbing them on his shirt, and then recite Asher Yatzar (Shulchan Aruch 4:22).