In all likelihood, this is about "designating" the examiner as suitable for an individual test. Details are in Information Notice 2012/156, found
here.
If you take this literally, prior permission is required for each skill test. Currently there are two options for a training organisation to obtain this permission - they can either get individual permission for each test, or blanket permission for a "panel" of examiners they normally use.
In theory, they could reject the skill test as invalid, but given that they have not rejected your application outright it does not look like they are minded to do so in your case.
I would lean on the training organisation to unblock this - it is not the examiner who needs to do the process, but BA flight training.
It happened to me two days ago (CAA same day counter service), it took a few hours but the ATO resolved this with the CAA; presumably some paperwork got lost or not captured in the system.