I hate to be the bearer of bad news, but unless your new supervisor/PI is willing to integrate your old work into their research program, you are probably just going to have to start over. Even if a new PI were willing to help, it would be risky unless they have sufficient background and knowledge in your previous work to properly evaluate it and give you guidance. You may just need to abandon (for now) that line of inquiry and integrate yourself and your research into a new working group if necessary.
There are a couple of points that you should keep in mind. First is that you have learned something, presumably, during that year and you haven't lost it. That might be something valuable in itself. Second, you can, perhaps, return to that line of research in the future, once you have finished your degree and established yourself.
It is important to note that it is common for doctoral students to "lose" time in unproductive research, though a full year might be somewhat rare. This is due to the very nature of research in many fields. There is no schedule on which you can depend for results. Many follow a line of research that proves unproductive and leads to abandonment and starting over with a new question. This is quite different from your situation, but has a similar result.
In my own case I worked on three problems in sequence. This was in mathematics. The first problem proved to be too easy. I could generate theorems at the rate of about one per day (maybe my memory of long ago is a bit faulty, but it was something like that). My advisor and I decided after a bit that this wasn't sufficiently "deep" to result in much of anything meaningful, so I abandoned it. The second problem was too hard and after several weeks of hard thought I couldn't get the smallest insight into a solution. I haven't followed up to see if anyone after me (50 or so years) has had a success with it. But, on advice of my advisor, I abandoned this one also. As with the typical Three Bears story, the third problem was "just right". I was hard and interesting and resulted in a nice, unified, theory, not just a few isolated theorems.
I don't think there was anything particularly unique about my path. Research is looking at the unknown. Scheduling when it can become "known" is essentially impossible. Even Einstein worked on Special Relativity for ten years before he came to the resolution.
In your case other considerations, not very nice, are forcing you to adapt to the real world, messy as it is. I suggest that you look forward, not back. Yes, it is painful, but if you treat it as just a learning experience and move on, you still have the opportunity, with a new PI, to start a successful career.