The TVTropes Trope Finder is where you can come to ask questions like "Do we have this one?" and "What's the trope about...?" Trying to rediscover a long lost show or other medium but need a little help? Head to Media Finder and try your luck there. Want to propose a new trope? You should be over at You Know, That Thing Where.
The hookup itself would be Pre-Climax Climax. That trope's the closest thing I can think of, even though it doesn't necessarily require cheating.
Looking more from the motivation side, if he loses his morals/inhibitions because "nothing matters anymore," then it could be Apocalypse Anarchy and will almost certainly be the result of a Despair Event Horizon. If it's not so much nihilistic ambivalence as it is a desire to live life to its fullest while he can, then Like You Were Dying could be a better fit.
If they keep going when the nuclear war actually *starts* because they're just past the point of caring, it's also While Rome Burns.
“Quitters never lose!”
Let's say a man decided to cheat on his fiancé with a random woman after hearing a nuclear war was very likely to happen. What trope would that fall under?