How can a university student reconcile attending medical lectures that may extend until late with performing the obligatory prayers (Dhuhr, Asr, Maghrib, Isha) on time, given the presence of a small prayer room next to the lecture hall?

There is no harm in praying in the university prayer room if going to the mosque would cause harm to one's studies.

It is not permissible to delay the prayer beyond its prescribed time due to studies. Rather, it must be performed on time, for prayer takes precedence over studies, and it is the first thing a servant will be held accountable for on the Day of Judgment.