I am so sorry to read about your Mother. My sincere condolences.
It sounds as though it has been a very tough few years for you and your family.
If you are not feeling the Imperial course, or London. Then don’t do it.
Nobody cares how many years you take between school and university, and it sounds as though you have very good grounds for the decisions that you have made. It also sounds from your existing A levels, and the fact that you have taught yourself A level maths in such a short period at such a difficult time, is seriously impressive. I am sure any university that you end up going to will be delighted to have you.
Imperial is a great uni, but personally I am not a fan of the whole London vibe, it seems very busy, and a bit impersonal - and as you say, it may not be quite what you need at the moment. Only you can decide what will work best for you in the circumstances. It is a lot to deal with in a short period.
If you do fancy medicine, or think, perhaps it is something you might come to return to more in the future, you could have a go at sitting the UCAT, with a bit of practice I think you would probably do very well. Plus you might relish the challenge. If you sit it this summer it leaves your options open for the year ahead. If you don’t, you can’t change your mind for another year. And if you did do well, then combined with your A level results there are plenty of med schools that only play lip service to GCSEs, including Cambridge. And that is without your extenuating circumstances.
Take some time for you. I understand your concerns for your family, but don’t neglect yourself in all this. A little mental fun like the UCAT as well as a bit of physical effort like the worlds toughest mudder sounds like an excellent idea. It’s been a hard couple of years, and you have achieved so much. There is no rush for the next stage. Good luck.