AFAIK home schooling is perfectly legal in DK, but almost unheard of. I can't recall knowing of someone who was home schooled for the past 30 years or so.
That would require either parents who are extremely qualified as teachers, as they have to teach accoriding to government standards and it would also be recommended against since children need inputs different from what they will get at a home school, where the environment tend to be very static. - And indeed they would be severely disadvantaged later on in an educational system that is very much based on social skills and group work.
Or that the parents employ skilled teachers, which would be insanely expensive!
The normal procedure if a child is not doing well, is for the school to sound the alarm to the parents. If that doesn't work, the authorities may be alerted, because there may be issues at home, which affects the learning.
The authorities may make investigations as to issues at home or with the child, if need be putting pressure on the parents and in extreme circumstances remove the child temporarlily or permanently from the parents.
That has happened when say immigrant parents take their children out for three months or so and travel on an extended holiday to the old country. - For several years.
Children can of course take a leave for a year, for say sailing around the world, that's after all up to the parents to decide. But it is expected that they recieve some basic schooling while away and the price can easily be for the child to leave his/her class and not move up one grade at the beginning of the school year. Nine years of basic schooling is after all mandatory in DK. (0 grade which Bella is attending is not counted, so for the overwhelming majority, basic schooling lasts ten years).