"Gone! Gone! Bless his little beak, he's gone! Musta pulled himself free! Beaky, yeh clever boy!"
-- Rubeus Hagrid
Harry and Hermione lead the hippogriff away while the execution party – an elderly member of the Committee for the Disposal of Dangerous Creatures, Walden Macnair (the executioner), Dumbledore, and Hagrid – is busy inside Hagrid’s hut. Prompted by Dumbledore, Harry and Hermione used Hermione’s Time-Turner to return to the time of the execution. They then use Buckbeak to reach Sirius Black, and Sirius and Buckbeak escape from Hogwarts together.
The official execution notice, which Fudge reads aloud, states that Buckbeak is to "be executed on the sixth of June at sundown - ". Additionally, Harry and Hermione travel three hours back from 11:55 PM. (PA21) This puts what happens just around 9:00 PM, which would be around sundown in early June.
Buckbeak's troubles were brought on by Lucius Malfoy, who wanted to get back at Hagrid for Draco's injury. In PA16, Hagrid said that the committee-appointed executioner, Walden Macnair, is an old pal of Lucius Malfoy's. In GF it is revealed that they are also fellow Death Eaters. -BB