In your sentence, "demanded" is in the past. If I want to say "now," I would say, "I demand that he apologize." "Should have apologized" is used to make a criticism or express regret about a past action. If I am talking to someone in the present about a past situation, I might say, "He should have apologized." If I want to tell a story about a time in the past when I was demanding that someone apologize, I would use your first sentence, "I demanded he should apologize." However, it would probably be more common to leave out "should" and say, "I demanded (that) he apologize.

Matthew has already brilliantly answered the question. Can i just add in English English we write it ‘apologise’!

He hasn’t apologized yet, so there is no need for the past tense. In fact it seems to me the best way to say this would be “I demanded that he apologize”.