I'll echo others in saying that if you deregister now, it may be difficult to find somewhere for your son to sit exams in spring 2022. At the same time, his mental health has to come first and if school is making him miserable, that could be a more important consideration than achieving exams at the "usual" age. There are no deadlines for education except that IIRC many college courses must be started before the age of 19 to qualify for funding, but that still gives a few years leeway.
He is not likely to achieve well academically although he is very driven and wants to be out working. He wants to go to college next year and should get what he needs for the course he wants to do.
Reassure him that there are other options if that doesn't happen. The college might take him onto a less advanced course in autumn 2022 so he could progress to his preferred course the following year. Our local college offers a year-long package of five "catch-up" GCSEs, for example. My eldest, who had no GCSEs, considered that option at the age of 17. Another possibility is to prepare for exams from home, but aim to sit them a bit later, for example there are some sittings in autumn or winter or he could do them in spring 2023.
He might use this year to start some part-time work alongside informal relaxed home education - the two of you can do whatever subjects seem appropriate. For example, he might keep the maths ticking over and also do practical life skills such as DIY, cookery, and money management. (I say "part-time" because any job this year would have to conform with child labour laws.)
From June 2022 he will be finished with Compulsory School Age, whereupon most restrictions on working are lifted. The duty to remain in education after Y11 is not being enforced and there are no penalties for not participating. So if he didn't want to go to college, he could go into full-time work instead. This was what my kid originally intended to do, knowing that they could go back and get qualifications later on.
With home education, the end of Y11 is not the cliff face which it is at school. Outside the school system, there is no particular reason why exams need to be done at that age. Your son and you may feel a little more relaxed if you remember that there are many backup plans available if he doesn't sit exams this year. School is absurdly inflexible and he's probably been getting the message for years that his life will be a disaster if he doesn't sit a good number of GCSEs, and achieve good results in them, at exactly the "right" age. It really isn't like that, unless he buys that story and becomes unduly discouraged about the direction his life is taking.