Is it permissible to prepare the cup for the next person waiting to wash for bread? If I have already washed, can I put the cup in the sink under the faucet that is already running to fill up for the next person? If the faucet is not running, can I turn it on and start the water to fill up the cup for the next person?
ב"ה Shalom Interestingly enough, the Shulchan Aruch (או"ח קס"ו :א) writes as a first opinion that one need not be cautious not to interrupt between washing and Hamotzi. This is because when the table is set and ready to be eaten from one is not easily distracted. The other opinion is, which is more commonly observed, that even idling between washing and Hamotzi should be avoided and of course not to speak even Divrei Torah. According to some, it is an act of piety –Midat Chasidut- not even to look at one's watch. In regard to your question, it would seem from the Mishna Brura, that filling the cup for the person after you wouldn’t be considered a distraction and especially if the person after you is eating with you it is probably a need for the se'uda which is permitted. see; שולחן ערוך אורח חיים, סימן קסו, סעיף א, משנה ברורה סימן קסו ס"ק ב, ילקוט יוסף נטילת ידים וברכות סימן קסו, פסקי תשובות. All the best