The epilogue to Harry Potter and the Deathly Hallows introduces several members of the next generation of the Weasley extended family. Not long after, J.K. Rowling released a family tree, including names not mentioned in the epilogue:
This family tree does not include any ages or dates of birth, however,
and gets Fred's date of death wrong, as he died in 1998.
Using the date of the epilogue (September 1, 2017) as a reference point, how old are each of the next-generation Weasley children?
There's some information available on HP Lexicon's page on the Weasleys:
Victoire Weasley (b. May 2, circa 1999 or 2000)
Hogwarts c. 2011. Victoire is the daughter of Bill and Fleur Weasley. Harry’s son, James, caught Victoire snogging Teddy Lupin on Platform 9 ¾ as she was preparing to leave for what was most likely her seventh year at Hogwarts; at the time she would probably have been seventeen, and Teddy nineteen (DH/e). She was born on May 2nd, the anniversary of the victory over Voldemort (YL).
Rose Weasley (b. circa 2006)
Rose is the older of Ron and Hermione’s two children, and their only daughter. On 1 September 2017, Rose was preparing to take the Hogwarts Express to her first year at school, and was rather nervous about whether she’d be sorted into Gryffindor (DH/e).
Hugo Weasley (b. circa 2006-2009)
The younger of Ron and Hermione’s two children, Hugo seems to be good friends with Lily Potter (DH/e).
In the epilogue to Harry Potter and the Deathly Hallows which takes place September of 2017, Hugo was not yet old enough to attend Hogwarts, but instead hangs out laughing with Lily on Platform 9 ¾ while his older sister, Rose, prepares to depart for her first year at Hogwarts. Though we don’t know his exact age, Lily Potter was nine at the time, and it’s likely the two of them were within a year or so of each other’s age. This would place Hugo’s birth roughly between 2006 and 2009, and his first year at Hogwarts at circa 2019 (DH/e).
And their page on the Potters:
James Sirius Potter (b. circa 2005)
The oldest son of Harry and Ginny Potter, named for Harry’s father. He seems to have inherited a bit of Weasley – interrupting his cousins while snogging (DH/e) – as well as a bit of his namesake, stealing the Marauder’s Map from his father’s desk (BLC).
Albus Severus Potter (b. 2006)
Middle child of Harry and Ginny Potter, named for Albus Dumbledore and Severus Snape. When he begins Hogwarts he’s quite nervous about being sorted into Slytherin, thanks in large part to the goading of his older brother James; Harry reassures him that the Sorting Hat will take his choice into account if he wants it to (DH/e).
Lily Luna Potter (b. 2008)
Youngest child and only daughter of Harry and Ginny Potter, named for Harry’s mother, Lily. Like her mother twenty-eight years earlier, Lily was jealous as she stood on Platform 9 ¾, watching her older brothers go off to Hogwarts (DH/e).
For information on the rest, I've had to resort to the notoriously unreliable HP Wikia:
Dominique Weasley (b. post-2000)
She must have been born post-2000, since she's younger than Victoire. -- source.
Louis Weasley (b. post-2000)
He must also have been born post-2000, since he's younger than Dominique.
Molly Weasley II (b. 2000-2006)
Percy´s family was present on Kings Cross in 2017. So, at least one of the children was probably ready to start at Hogwarts. Molly, being the elder of Percy's offspring, may have been the same age as Albus - 11 - or may have been older; however, she cannot be born any earlier than 3rd May 2000 (as Victoire was born the day before, on May 2nd, 2000, and is the oldest) and cannot be born any later than August 31st, 2006 (as this would make her too young to be attending Hogwarts). -- source
Lucy Weasley (b. post-2000)
She must have been born post-2000, since she's younger than Molly.
Fred Weasley II (b. post-2000)
Victoire is said to be the "eldest" of her Weasley cousins and to be attending Hogwarts in 2017, she cannot have been born before 2000. Thus, to be younger than her, Fred must have been born some time after May 2nd, 2000. -- source
Roxanne Weasley (b. post-2000)
She must have been born post-2000, since she's younger than Fred.
That covers everyone, and it seems to be all the information currently available. Naturally we know more about the children of the main heroes (Harry, Ginny, Ron, and Hermione) than about their assorted cousins. This article from the Times may possibly go into further detail, but I'm unable to view it without opening my wallet - if anyone has already subscribed to view articles on the Times website, please leave a comment to let me know of anything relevant I can add, or post a competing answer if there's enough extra information there to warrant it.
Rose-Granger Weasley
Probably eleven, since she is wearing "brand-new Hogwarts robes" as of the epilogue:
“Hi,” said Albus, sounding immensely relieved.
Rose, who was already wearing her brand-new Hogwarts robes, beamed at him.
Albus Severus Potter
Clearly he is eleven, since he is going off to Hogwarts.
James Sirius Potter
James went to Hogwarts the previous year.
“We wrote to James three times a week last year,” said Ginny.
However, it was not his first, since he first attended in Fall 2015. As such, he is thirteen.
Lily Luna Potter
She went to Hogwarts two years after Albus Severus:
“It won’t be long now, and you’ll be going too,” Harry told her.
“Two years,” sniffed Lily. “I want to go now!”
As such, she is nine at the time of the epilogue.
Hugo Granger-Weasley
We don't really know. He must be younger than eleven, though, since he cannot go to Hogwarts as of the epilogue:
Back on the platform, they found Lily and Hugo, Rose’s younger brother, having an animated discussion about which House they would be sorted into when they finally went to Hogwarts.
He is probably similar in age to Lily Luna.
Fred Weasley Jr.
We don't really know, but certainly less than nineteen, since he could not have been conceived before the Battle of Hogwarts.
Roxanne Weasley
Same, we do not really know.
Molly Weasley Jr.
Same
Lucy Weasley
Same
Victoire Weasley
“Teddy’s back there,” he said breathlessly, pointing back over his shoulder into the billowing clouds of steam. “Just seen him! And guess what he’s doing? Snogging Victoire!”
She is most likely at Hogwarts, and thus is seventeen or younger.
Dominique Weasley
Younger than Victoire, so less than seventeen.
Louis Weasley
Same.
