Kaoru Mori has not given exact ages to the characters (except to Ilse whose fifth birthday was featured), but instead, she gave the age order of the characters in the extra material of the ninth manga volume. Using this and other hints in the series, let's try to figure out their ages! And here comes the age list, from youngest to oldest. If they are on the same line, they equally old. You can add comments about them under the names! (The troper has only Finnish versions of Victorian Romance Emma manga volumes and thus she is does not know the English translation. Feel free to make corrections.)
Theo the Squirrel
He is still alive after the time skip. The life span of a Gray squirrel is approximately six years.
Unless Erich got another squirrel and named it Theo, too.
Her fifth birthday was featured in Chapter 19, in the third manga volume.
After the time skip, she is still clearly a round-faced child.
Nini the Monkey
He/She is older than Ilse and is still alive after time skip.
His mom asked in a letter of hers, "Can Colin write already?" So he may be around 6 or 7.
After the time skip, he is still a child to my eyes.
In a forest trip in Extra Story Chapter 7, a maid (Adele) escorted him when he had to pee.
He also cried aloud when Theo went missing.
Extra Story Chapter 8 states that Erich's parents Wilhelm and Dorothea have been married for eight and half years. Giving the time period and their social status, it can be safely assumed that Dorothea was not pregnant with Erich when they were married. That would make Erich at most 7 years and 9 months old and at least 5 years and 9 months old due to his sister's age.
After the time skip, a border-line teenager? He has got ganglier limbs but not much shoulders.
Vivian "Vivi" Jones
A bratty child before the time skip.
After the time skip, she has clearly reached puberty: her breasts have started to grow, and she mentions she had grown two inches since last year. Also, Grace forbids her from drinking alcohol, against which Vivi insists she is an adult already, and against that Grace tells that Vivi definitely isn't an adult yet.
Arthur Jones & Jan
They look like teenagers before the time skip (Arthur is even shorter than Grace, but that may be just that Teens Are Short).
They look like young men after the time skip.
Arthur is studying at a boarding school at Eton and he's about to graduate there. Students there are 13-18 years old, making Arthur perhaps 17-18 years old.
The youngest students there, featured in Exta Story Chapter 15, can't be 13 years old to this troper's eyes!
After the time skip, Arthur is "still in the college/university", and reading between the lines, it appears that he is old enough to drink alcohol without anyone complaining (he talks about how he isn't into drinking alcohol and how alcohol belongs more into the society than into academics).
Old enough to be engaged in Victorian England.
She is not fully grown physically, as in the beach chapter in Extra Stories (yes, there is one, completed with authentic Victorian swimming dresses), she complains about how thin arms she has, and her maid Annie replies that she is just so young.
Eleanor and Grace's three nameless gossiping lady friends
A comic strips in Extra Stories reveal that she has been mistaken into Colin's mother at least once, though most likely she isn't literally old enough to be his mother (she was a child when their mother Aurelia left the family to retire on countryside and Colin had already been born that time).
Hakim Atawari & Emma
In his and William's childhood flashback (Extra Story Chapter 9), Hakim was maybe 2-4 inches shorter to William, so they can't have veryveryvery large age gap. I'd say William is 1-3 years older to Hakim. Plus, Indians are typically shorter to caucasians: also when adult, Hakim is slightly shorter to William (see Chapter 6, the library scene, when Emma talks to a librarian and Hakim and William browse the book shelves).
Emma was 15 when she was employed by Kelly (Chapter 10, see more in Known dates in the series). Given that Kelly employed her after she had quit working as a home teacher, Emma must be less than 25 years old in the present time.
William Jones & Robert
Kelly was William's home teacher for four years when he was a child (roughly ten years ago as Kelly states) and she was still working as a home teacher (she was possibly around her 50s, see her notes). In the first manga chapter, Kelly jokes about that if William is still wetting his bed, hinting William's age that time.
Alma & Hans
In the age lineup, she comments, "I am still rather young." However, she is also Ilse and Erich's mother.
In the age lineup, he says, "I am rather young, too." However, he is also Ilse and Erich's father.
Extra Story Chapter 8 states that Wilhelm and Dorothea have been married for eight and half years. Wilhelm also comments that around the time when he proposed Dorothea, he was old enough to grow mustache without looking silly. That might make him 20-something years old that time and currently on his late twenties or on his thirties.
In Aurelia and Richard's flashback, Richard didn't look a lot older to Aurelia.
Final epilogue in manga volume 10: Unless Maria is joking when calling Johanna "mother", she is Johanna's daughter.
Kelly Stownar & Al
During the Great Exhibition in Crystal Palace in 1851, she was 18-19 years old: The extra story of her and her husband Douglas' visit there (Extra Story Chapter 1-2) happened a year before he fell fatally ill and died without seeing the next summer. Kelly was married in age of 18 and widowed in age of 20 (Chapter 5). So she was born in 1832 or 1833.
After that, it was stated she had worked as a home teacher "over 30 years". She retired after that, though we have no information about how long she has been retired (except that she has taught William when he was a child). So, it makes her at least 50 years old and possibly closer to 60 old.
Given that the series is set to the end of 19th century (or mid-to-late 1890s?), I guess she could be on her 60s.
Busted! Her grave stone in Volume 3 claims that she was born in 1838. However, she doesn't really look like 13 years old in the Extra Story... maybe Kaoru Mori changed her mind?
In omake comics sections, Stevens comments that he assumes he won't live long enough to see Colin growing adult.
In addition to that, Adele's age is unknown and Monica doesn't want to tell her age.
Length of the time skip
Another question: How long is the time skip between the end of manga volume 7 (the end of the actual series) and the final epilogue of the series in the volume 10, or William and Emma's wedding? Some notes:
The start of the series is set at "the end of the 19th century", and we have the Victorian era here; The epilogue is set at "the start of the 20th century", and we have the Edwardian era here.
Queen Victoria died in January 1901, so this was the switch point between the Victorian and the Edwardian eras.
Theo and Nini are still alive after the time skip: according to The Other Wiki, at least the life span of a Gray squirrel is approximately six years.
Unless Erich got another squirrel and named it Theo, too.
During the time skip, Grace had got married and given birth to a son who is "soon six months old" during William and Emma's wedding. Also, William's conversation with Bill the Gardener hinted that Lionel and Grace's wedding was set to late spring or summer, and it looks like there is late spring or summer after the time skip, too. This troper's guess is that there has been at least almost 6 months (son's age) + 9 months (pregnancy) = almost 15 months when Lionel and Grace's wedding was held, and when this is rounded to whole years, it may be possible that their wedding was held at least two years before William and Emma's one. Unless Grace was pregnant during her wedding.
Children and teens have clearly aged (see earlier), while no notable appearance changes have happened to adults.
Known dates in the series (to support two earlier questions)
Here I list some more or less exact dates that are shown in the series or RL historical dates:
1832-1833: Kelly Stownar was born (if you believe the Extra Story in Volume 8 and Kelly's narration of her own life in Volume 1, see Kelly's notes in the age question).
1838: Kelly Stownar was born (if you believe the grave stone in Volume 3, see Kelly's notes in the age question).
1851: The Great Exhibition (events of Extra Story Chapters 1 and 2)
1852-1853: Kelly's husband Douglas fell fatally ill a year after the Great Exhibition and died "without seeing the next summer".
1890s: Kelly's death year; the events of Victorian Romance Emma. (The numbers 1, 8 and 9 are visible on Kelly's grave stone, but the final number is blurry.)
October 8, 1896: The 5th Extra Story Chapter ("The Times") is basically about people reading (and using) The Times around that time, judging by the date on the newspaper on the cover page.
Judging by where Violet appears, who is with her and what they are talking about in this chapter, most likely this chapter is set after the events of main series but not long after it. So have we found the event year of Victorian Romance Emma?
1901: Queen Victoria died: switch between Victorian and Edwardian eras and thus also a year included into the time skip.
In addition to these, we know a few related events:
Preteen!Emma was hired as a maid for Kelly when Kelly was about to retire from her home teacher job: This had to happen in 1880s, as Kelly has stated she worked as a home teacher for "over 30 years" since her husband died in 1850s.
She was not preteen but 15 years old when Kelly hired her. That was stated in manga Chapter 10.
Prior that, Kid!William had been taught by Kelly for four years roughly a decade ago (in Chapter 1, Kelly states that William "has not changed in ten years").
When the Jones Kids' mother Aurelia left her family to retire to countryside, Colin was an infant, Vivi was a toddler, Arthur was wearing a sailor suit typical for children, Grace was still clearly a child, and William appeared like a preteen.
Ilse and Erich's English skills
How was it possible that Ilse and Erich, who apparently were unable to speak anything else than their native language German before the time skip, speak apparently at least quite good English after the time skip? (They chat with Colin and Vivi respectively and apparently have no big language barriers. Though they don't talk about very complicated issues.) Mad English studying motivated by the contacts to Joneses? Well, also depends on how long the time skip was. And of course, children can be amazing what comes to learning new things.