I have no idea if this is possible in UK, but if you take a longer and wider view you can probably find success. However you might have to split your quest into two separate ones. As I noted in a comment, the PhD and the specific research are separate things and can be done independently.
My advice is to prioritize the PhD over the specific research topic at the current time.
The problem with starting a degree and bringing your research with you is that you need to find a compatible advisor as well. This makes the "gate" much narrower. If the research comes with funding then it might work, but putting conditions on an acceptance lessens the chance of success. It is uncommon enough to be a problem.
I think you might have been abused by your original institution, though. If your professor left some accommodation should have been made. The tag is expulsion, but I'm going to assume that termination is a better descriptive. Some programs are time limited, though, again, something should have been done if your professor left. I'm assuming, of course, that you weren't actually expelled for some unethical practice, but just that the clock ran out or you didn't pass some exams due to various issues, whether related or not.
Let me suggest that you first explore getting into a PhD program somewhere. The US might be a more understanding option for you if you can put together a good application. People here change institutions with some regularity and for various reasons. The main issue might be finding letters of recommendation, which are relatively important. Perhaps your old professor would vouch for you. Don't insist on continuing the research you've started, though you can suggest it, but get into a department in a closely aligned field. If you get accepted, perhaps you can work out the research issue, but if you insist on that as a condition, you have fewer chances.
If you complete the doctorate, with a dissertation in that related field, then you will be able to develop a career in which you have a safe space in which to return to the original research. Don't let that topic become a block on getting a degree, however. If the topic is not a "hot" one, as you suggest it isn't, then you are unlikely to get scooped in the meantime and your current situation isn't conducive to completing it in any case. Get to a better place in which you can make your own decisions.
Two additional thoughts, though neither might be possible. Your old professor really owes you something for abandoning you. It's a mistake to bring that up, but in some cases a student will move with the professor to the new institution. Maybe that is worth looking into. You probably need to contact them in any case for a letter.
The second thought is that the original institution probably also owes you and it might be worth exploring whether you can return there. Your situation (family...) has changed, and some things were out of your control at the time.
Alternatively, if the research is more important to you than a degree and career, then you can possibly do the research as an independent researcher and find a publisher. If you can leverage that into a doctoral program admission then you can wind up ok, but I think that is the (much) riskier path.