Admiral - premium change (increase) after claim on policy

Next thing they'll be voiding the policy because when you took it out you didn't declare that you were going to have an accident.

Took out a policy, a few days later I had a 'an at fault' accident - that's it as far as I can see.  Damage was to third party, no damage to my vehicle,

Have you actually called them and asked for an explanation yet? They are probably better placed to provide the reason than any of us here.

I did get to the bottom of it ... it was a claim my wife had on her policy in August '21, and because I had added her as a named driver (given she's about to be my ex-wife this was silly) they said I had to declare her claim (which I didn't know anything about) ... anyway, end result was to remove her as a named driver, and I ended up getting a smallish refund 

Every single website I've been on to get insurance has asked the question "Does this driver have any accidents or claims within the past X years?"... which one didn't ask?!

I am sure they all ask. However that's not completely the point.

Given OP was unaware his belief of the accuracy of his answers fulfilled the declaration he made.

I had a similar situation with home/buildings insurance about 20 years ago.  I made a claim for a new garage door after a break-in.  Subsequently, I changed insurers and the new insurer did not ask me if I had made any claims.  A couple of years later had to make a claim with the new insurer for replacing a sewer pipe because of tree-root damage.  I remember receiving a phone call one morning from the insurer telling me that I had failed to declare the claim from a couple of years earlier and that they could not settle the new claim until my premium was adjusted.  If I recall correctly, the additional premium was about £30-40 so I just paid it.

Looking back, I am fairly certain that the insurer was on shaky ground.  Why else would they offer simply to accept a payment that reflected the difference in historical premium?  I could have argued it, but it wasn't worth the time and energy.