As a non-native English speaker myself, I advise you not to try to understand the English translation, specially if English is not your native language. I think it's more useful to just understand the situation in which you can use each expression, so you don't need to rely on other languages to understand those expressions.
Let's imagine you are appointed at 5pm and that you should be in that room by 5pm, and let's compare each case.
I think that saying 遅れた場合は, it is conveyed that you actually arrived to the room, but you did it at a later time than it should be (in our example, this means the case where you arrived at the room but it's already 5:15, for instance). So it would be "in the case you had arrived late to the room".
If you said 遅れている場合は, it would be "in the case you are running late", so in this scenario you would be travelling from somewhere to the room, and late on time, but you would still have a chance to catch up and arrive on time by running. In our example, you are in the other side of the city and you are on your way to the room. However, it's already 4:45, so it is very unlikely that you will arrive in time. But you are not in the room yet, and it's not 5:00pm yet.
If you say 遅れる場合は, it sounds like "in the case you are to be late", or "in the case you will arrive late".
My English translations are probably wrong grammarwise, but I hope you got the point.
It might help you to understand it if you think of the expression Verbた + 時 and Verb-辞書形 + 時, where a similar thing happens:
家に帰った時、「ただいま」と言います。"We say "tadaima" when we have come back home". You are already at home.
家に帰る時、「お疲れ様です」と言います。"We say "otsukaresamadesu" when we go back home". You are not at home yet. You are about to go back, but you are still in the workplace.