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Do I have to submit an attendance register and do I need to keep a record of time spent on each subject?

No not at all. A register of attendance is for school requirements. You do not need to give anyone a register to show that your children are at home. Your curriculum and the work your children are doing at home will be the evidence of the learning which has taken place. Your child will be working through workbooks which will show the results of their learning due to the fact that there are built in assessments, test and exams in the Oikos resources/curricula.