Year 11 is an undesirable time to leave school, if the young person is tolerably happy, progressing reasonably well, and has the expectation of achieving what he wants to at school. Whatever work your son may have done last year towards GCSEs probably would not count toward exams he could sit while being home educated though obviously if the school has taught him anything worth learning rather than just teaching to the test, Y10 cannot be considered a write-off as he will have acquired lots of transferable knowledge .
HOWEVER, you've described a very unhappy, anxious, ill young man who is not getting the support he needs at school, and onto whom the school is about to pile even more pressure. It doesn't look promising, does it? I assume you are not considering home ed all of a sudden based on one day in Y11, but that the situation has been pretty bad for some time now.
With home education, there is no hurry to achieve a certain number of exams at a certain age. So... what if your son does end up starting over on different exams? What if you struggle to find exam centres and he doesn't manage to get the qualifications he was hoping for this year?
Then he'll have some time this year to recover his confidence, have a break, and perhaps even improve his health a little. He can explore whatever truly interests him, dabble in different areas, broaden his education in topics not permitted at school. He can do some exams next year, or the year after. He can do them through distance learning, or with you, or with a tutor, or at college. He can do them as and when he feels up to it and has mastered the material and is motivated to do them. He can do them a few at a time or all at once. He can do many or few, or none at all if he sees an alternative way forward which doesn't depend on exams. He doesn't have to do any subjects which aren't necessary to his future.
You've been on the school treadmill for many years, moving relentlessly toward a fixed point: end of Y11, GCSEs, end of school's opportunity to squeeze as much performance out of your son as possible while they still have him under their control. They are panicking, because they have targets and his ill-health jeopardises their results. You don't have to panic. You don't have a deadline. Your only target lies years in the future: a well-rounded, happy, educated young man. Step back and look at your lad's whole future. It doesn't have to be like this. First make him happy, then take some time to look around and see how he wants to move forward.