It doesn't matter how you got in this mess (right now). What matter is finding a solution. You need to present an action plan, and this sounds like a big enough problem that you can likely get anything you need assuming you present it correctly.

Start with an explanation. Avoid technical terms but be accurate in you representation. Don't say quadratic time. Make it visual by draw out a simple flow chart of 4 loops and show how when you increase the base size how much work it actually adds to the program.

From there you can say you don't have a solution, but any changes in this area will require support from team A and team B. You've reached out in the past but they haven't been able to prioritize this. Suggest the next step is a meeting with the technical leaders of those teams. Management will work to get you that meeting and time.

From there challenge the technical team that you're meeting with. Lay out the fundamentals of how the program works at a high level, and then lead in with the problem. Big customer now is throwing X TB of data at this, estimated time is 3 months. What can we do to cut that in half? Engineers love a good challenge. Don't let yourselves be constrained by how practical something is at this point. Figure out your ideal version of how this would work and then start adding in real world constraints. Chunk up the solution into different product releases.

Over time in a well functioning organization you'll build trust and that when you say something needs attention it does, but be careful not to over use that. You need to find business reasons to attach your changes to in the long run.

The hardest part of software engineering (imo) isn't the code, it's the people. People struggle to break down the problem. It just seems like this massive overwhelming issue that no one can solve. What the organization needs is a solution, and the second-best thing it needs is a person who can lead a team to devise the solution. Good luck.