"….is this the kind of diplomatic neutrality one might expect of a ruthless careerist whose long-term ambition is undoubtedly to be Minister for Magic?"
-- Rita Skeeter (Pm)
By the time that the Hogwarts Express is leaving King’s Cross for the start of her daughter Rose’s fourth year, Hermione Granger has become the Minister for Magic (CC1.5).
This had been predicted by Rita Skeeter during her commentary at the 2014 Quidditch World Cup final (Pm).
This event occurs in the main "through" timeline of Harry Potter and the Cursed Child. Use of a Time-Turner by Albus Potter and Scorpius Malfoy cancels this event in an alternate timeline.
The exact date is unknown, but it will have occurred prior to Summer 2020.
From the Web
The theatre production opened in the West End of London at the Palace Theatre on 30 July 2016, with Noma Dumezweni as Hermione Granger: Harry Potter and the Cursed Child, Parts One and Two
Screenshots of the Daily Prophet's coverage of the 2014 Quidditch World Cup: http://imgur.com/a/AXutv