Not necessarily. They announced their plan to move to Sweden in March, and they cancelled their plan in late June. Perhaps they assessed the children's schooling at the end of the school year and changed their minds.
Things can change a lot from March to June.
Somebody was responding to the example you gave in your original post:
I was just thinking about whether next year would be a more natural year to transfer the kids to a new school without too much disruption. Based on their ages, I would guess that Leonore, who is nine, is probably in fourth grade. Nicolas, who is eight, is probably in third grade. (Give or take a year, depending on what age they started.)
Neither would be likely to change schools for another few years here in the United States, but I wonder if next year might the age that Swedish students change schools.
The age at which children normally transfer schools in Sweden did not change between March and June 2023, and thus is something that the parents should, and probably did, already assess before March.
To answer your original question:
"In the Swedish school system, children go to school for at least ten years from the year they turn six, as mandated by the Swedish Education Act [...] Swedish compulsory schooling consists of four stages: förskoleklass (‘preschool year’ or year 0), lågstadiet (years 1–3), mellanstadiet (years 4–6) and högstadiet (years 7–9). Children between ages six and thirteen are also offered out-of-school care before and after school hours."
The Swedish school system | sweden.se
Campus Manilla, the school which Princess Estelle and Prince Oscar attend and which Princess Madeleine was rumored to be considering for her children, includes a primary school which covers years 1-9 (i.e., children ages 6-16).
Grundskola - Campus Manilla
I went back and looked at the reporting in this thread of what the spokesperson for the royal house said, and the comment was that they weren't ready. That certainly would cover many scenarios, from the kids' schools to Chris getting his business activities arranged to selling their house here to getting a place in Sweden ready to move in.
Who knows?
The last two scenarios are unlikely. Court spokeswoman Margareta Thorgren
denied that the postponement was related to the home sale (or to immigration issues), and nothing has happened (as far as we know) between March and June to the couple's expanded and renovated (and rent-free) apartment at the Royal Mews. And while the children aren't necessarily guaranteed placements at their parents' first-choice school, schooling is free, compulsory and guaranteed in Sweden, so there is no concern about finding them a place.
He who pays the piper calls the tune. I have no doubt in Sweden that will be the in laws. Housing, schools, thats a lot of money. Not sure Chris is a breadwinner of that scale. The in laws may even pay for that already and in Sweden, Chris would be totally trapped by the in law family and the media.
Their housing and schooling would both be state-funded in Sweden: the former owing to the king's royal right of disposal over numerous state-owned residences, the latter being national policy.
Given how far down they are in the line to the throne, I don't understand why it is so important to Madeleine that her children have succession rights.
What leads you to think it is important to her? Besides, we have seen how for many people it is "so important" to have hundreds or thousands of people in line to the British throne; keeping numbers 9-11 in line seems much more reasonable in comparison.
Also, we don’t know, maybe their mother has only spoken to them in Swedish?
According to the court spokeswoman, the children are bilingual in Swedish and English.
In Hänt Magazine:
Madeleine's children have lived in the USA almost all their lives, and all three are kind of hidden royalty for many here in Sweden. You can then wonder about is how it really is with the Swedish language for the children. In an Instagram post that the royal family recently shared, several people wondered if the children spoke or understood Swedish.
But now the royal family reveal to Hänt how their language skills actually are.
"The children are bilingual", reveals Margareta Thorgren.
The princess' children can thus speak and understand both English and Swedish.
Sanningen om prinsessan Madeleines barn – efter frågorna _ Hänt