A note on the Time-Turning Machine: it looks like a giant Time Turner, but with seats. Lupin finds it quite a nauseating ride, and I don't blame him.
(The post in this world is delivered by mechanical owls run by the Royal Mail. In Wales, where the boys are here, this is called Post Brenhinowl. This is an excellent pun when you know that the Welsh for Royal Mail is Post Brenhinol, I promise)
What happens next? Up to you.
My personal fan theory is that they meet up with some other chaps who were (fictionally) there at the time, as below.
That's the Comte de la Fère, M. de Valon de Bracieux de Pierrefonds, and the Abbé D'Herblay (they have other names *wink*).