When you use the non-past (a.k.a present) tense to say something was going to happen, you, as the speaker, is standing at a reference point in the past and looking towards the future from that point. That's what the perspective in relative tense is like. The time expression should also be relative to the same reference point, if not absolute.
次の週 is not "next week" with no article but "the next week" or "the following week" relative to a past reference point.
Whether 月曜日 is "absolute" is debatable, but it's certainly different from expressions like 昨日 and 先週 as it directly points to a specific date (within what you might call a "reference window" centering around either reference point, though). If the listener doesn't know which Monday, that's another issue.
Time expressions like 昨日 and 先週, on the other hand, are always relative to the time of the utterance. If the time expression refers to the past, you are looking back from that point. The tense of the verb should also be matching this perspective. That means only your second sentence is correct.
But a little problem arises here. What if the students couldn't take the exam? We cannot say 受けた in that case.
Well, you still need to use the past tense, somehow.