Designing a PCB layout without having a schematic diagram (or netlist)?

It is technically possible to design a PCB layout without first completing a schematic, but extremely unlikely. It just makes it unnecessarily complicated. I think whoever you hired is not being fully honest with you.

I think a likely alternate explanation is that the company you hired turned around and outsourced the project to yet another company! But they may want you to believe they are doing the work in house, so they're keeping you in the dark and coming up with weird excuses.

It's also possible that they are doing the work in house, but underestimated the skill level necessary to complete a board of that complexity (it turns out that high-speed signals are hard to design for). So they're stringing you along until they figure it out.

Either way, your prospects don't look good. I would think very seriously about trying to get your money back and go with a company that can keep their promises.