The first application that presents itself, perhaps obviously, is to run WTWD as a prelude to a post-apoc RPG like Apocalypse World or etc. The game was in fact designed to function as a collaborative campaign generator in that sense. It can be played solo, or with any number of players. Depending on the players' approaches, methodological preferences and in-play ideas, it can be seen as anything from a silly one-shot game to a structured brainstorming device to a collaborative system for generating entire post-apocalyptic pre-game histories.
When played as a preface to an AW campaign, WTWD assumes the position of "Session Zero". You do it with the same people who are going to be players in the campaign. It provides an interesting mood-setter or intro/exposition piece and personally I found it is a great icebreaker for RPG newbies. But more importantly it leaves behind a physical artifact - the timeline of the fall - which exists outside of game space, but can be used as a practical tool by both players and MCs. This timeline assists in the spontaneous fabrication of character details which are a-priori faithful to the fictional world the group has collaboratively created, and it provides prompts and "common knowledge" for both players and MCs as regarding "what's possible to find out there". (The existence of casualwear body armor and survivalist caches in the game's sample apocalypse are good examples of this.)
On an AW-hack level, WTWD is an attempt to address some ramp-up problems I've sometimes seen in first sessions, especially when dealing with newer players or players who are accustomed to more structured CharGen procedures. The timeline will jar ideas on many levels as Session 1 is beginning: What destroyed the old world? What character class do I want to play? Which playbooks wouldn't really make sense in this world? What sorts of accoutrements might I have? What part of the world do we want to set the campaign in? What do the ruins of the destroyed world look like? What type of hazards have people learned to avoid? What do the old codgers sit around talking about? What did my father teach me? What sorts of things do I know are out there, perhaps numerous, perhaps rare, but nevertheless fairly common knowledge to anyone who's spent a decade or two growing up on this planet? As always, it is up to each gaming group to decide for themselves what sort of knowledge is known by the PCs and what's strictly "behind the scenes".
Technically some might say that this breaks the rule of "No Prep" but since the generation of broad planetary/environmental/historical details is already a collaborative experience belonging to Session 1, and since WTWD is equally collaborative and typically would involve all the same players, all it really does is (a) portion out a particular set of those questions to occur ahead of the more direct questions about character definition, and (b) provide an amusing brainstorming device to help you answer them together.