Thanks One Cal, I phoned them (AXA) 10 days ago because there has been a deafening silence on their part. That's when I got the info about the Green Card. I have since written to them with my concerns and laying out again all the facts as I see them.
What I don't get is that I've paid them their premium, made a legitimate claim (No Fault) and yet I'm still waiting for reimbursement. It shouldn't matter that they are in dispute with the third party.
I have provided a European Accident Statement (albeit in French) and I have an email from them stating that it is a No Fault claim.
I am totally fed up with the whole process now that I'm aware that a Green Card was not required.