Is what the student did by delaying the Dhuhr and Asr prayers until returning home to perform Ghusl considered a necessity due to a wet dream and the unavailability of warm water in the university mosque?

If you are unable to perform ghusl due to the coldness of the water and the inability to heat it or go home, then it is obligatory for you to perform tayammum and pray for each prayer time, according to the majority of scholars. However, according to the Shafi'is, you perform wudu' and wash as much of your body as you can, then perform tayammum and pray for each prayer time. This is the more cautious approach. It is forbidden to delay the prayer until its time has expired, and if you delay it, you must make it up, along with repentance and seeking forgiveness after performing ghusl from janabah.