I think that Philip stays longer, so that he can arrange Outremer affairs to his own satisfaction. In particular, two countries need a king.
Guy of Lusignan has no powerful allies left, so Conrad of Montferrat becomes king of Jerusalem immediately, with Philip's support. We don't know for sure whether Richard had a hand in Conrad's real-life assassination, so we don't know if it happens in this timeline too.
Cyprus might get a different king, too. Richard held on to Cyprus for about a year, then sold it to the Templars, who in turn sold it to ex-king Guy of Jerusalem after a short tenure. Leopold of Austria was related to the deposed rebel emperor Isaac Comnenus, and if his opinion carried any weight, Isaac might have been restored. Otherwise, it goes to a French candidate.
Anybody know who was second-in-command of the English forces?