It's likely only to be one of the children, though, not both, unless they have a positive test (in which case, the whole family has to self-isolate). Otherwise, it's only the "bubble" child who has to isolate. So if one is isolating, the other one can still move between households and go to school and childcare. Illogical, but isn't that the rules?
If I were mum and I had no annual leave and couldn't take unpaid leave so the likelihood was I'd lose my job, I'd probably organise for a friend or family member in my bubble (though not grandparents) to come and sit in my house. The self-isolating child should stay in their room and avoid contact with that person as far as possible (which this child sounds old enough to do if given unlimited computer games, snacks and meals).
It says this online about the guidance:
According to the government’s Track and Trace guidance, other members of the household do not need to self-isolate if a child has been sent home from school. It states: “If you live with other people, they do not need to self-isolate, but they should avoid contact with you as far as possible and follow advice on hygiene.”
Though, as many have pointed out, in the case of young children, it will likely be difficult for family members to avoid contact.
This rule also means, in practice, that siblings or children from the same household who go to the same school might still need to go to school, despite someone in their household having been sent home to self-isolate.