What is the ruling of Islamic law regarding a student remaining in school without going to the restroom when needed, due to the unavailability of water, considering the impact this has on him and his occasional need to go and wash himself upon returning home?

There is no need to hold urine or stool, as this is harmful to one's health. There is no harm in delaying Istinja' (cleansing oneself after defecating or urinating) except when intending to pray. Cleansing with paper is permissible if available, but it is not sufficient instead of water for a woman's urine according to some scholars, such as the Malikis, because it extends to the area of the buttocks.