All of them mean "at the end of" and they are sometimes interchangeable as long as they represent "at a point on and after an event or doing something" Among those three sentences, "五月の末に" (at the end of May or on the last day of May) apparently has a different definition and therefore cannot be replaced by others. Although those three expressions are close to one another, "のちに" is rather a plain expression for "after something", "末に" tends to emphasize that the following part is an eventual outcome, and "あげくに" tends to imply ending up with something unpleasant, unfavorable, or disappointing. (In this context, あげくに is often associated with an adverb さんざん or emphasized by another similar expression あげくの果てに).

Here are some examples to support above answers. 「うちの祖母は、留学中に知り合ったアメリカ人と大恋愛の末、駆け落ち同様に結婚したらしい。」 You can replace this 末with のちに, but it makes the story far less dramatic. 保守的な母「うちの娘、一人で勝手に留学に行って何年も帰ってこないかと思えば、あげくに外国人と結婚するだなんて何考えてるのかしら。」 For this one, あげくに is the most appropriate because the daughter's marrying with a foreigner is unpleasant outcome for the conservative mother.

I wouldnt say all those are meaning the same. ～のちに　"after a while" or "after a particular time" ～末に "end of" ～あげくに　"after all" (this one is particularly used for a consequence after happening a bad thing)