You are only required to continue to study. You are not forced to keep retaking the exam it's just a better option as if you finally manage to get the grade, you don't have to continue with the lessons. So if a person was homeschooled, didn't pass it the first time, continued to study but did not take any further exams, there's nothing against it.
"Pupils aged 16 will no longer be able to drop maths or English unless they have achieved at least a grade C in their GCSEs.
However, Matthew Hancock, the skills minister, said that there pupils will not be forced to take further tests in the subjects. He said there will only be a “high expectation” that teens forced to continue studying the subjects will go on to resit exams."