What is the ruling on delaying prayer past its prescribed time due to a lecture or an exam, and what should be done when it is embarrassing to ask to leave for prayer, or when one is prevented from doing so?

If you are able to perform the prayer within its prescribed time, then it is not obligatory to leave at the very beginning of the time, and it is permissible to perform it at any part of it. However, if the lecture or exam occupies the entire time, then you must perform the prayer or seek permission to perform it, and do not delay it until its time expires. If this is not possible, and you would be harmed by leaving, and the prayer is one that can be combined with the prayer before it or after it, then it is hoped that you would have an excuse for combining the two prayers.