They're a charity and a private institution, not a legal arm of the UK, an oversight organization of the EU, or a division of the UN. They have as much legal authority as any other private institution.

Not to say that it wouldn't be a good idea for AI to have the privilege of oversight of any court proceeding in the UK. But I believe such a thing would have to start as law from the Houses, not the discretion of an individual judge. If there is precedent to the contrary, I will stand corrected.