The problem is that the exam centre acted as a middleman between you and the examination board, passing your payment onto that board (although they might have added their own fee on top for actually providing you with a venue at which to take the exam). The exam centre will only have received a quarter of the examination fee back from the board
https://schoolsweek.co.uk/boards-return-just-25-of-fees-for-cancelled-exams/
so they're not in a position to refund the full fee to you.
You need to get a breakdown of the amount that you paid, so that you know how much was actually going to the exam centre itself and how much was going to the examination board. (You should be able to find out what the actual examination fee was by simply looking at the board's website. You can then subtract that from your payment to see how much the exam centre was charging you for providing the facility to sit the exam).
You should then be able to argue for the repayment of the 'facility' fee (since it was never provided to you) and for the quarter of the actual examination fee that the centre received back from the board (since the centre was acting as your agent when it transferred that element of your payment to the board).
However I can't see any way that you will be able to get a refund of the entire payment, unless someone makes a successful legal challenge to the exam board's decision to only refund a quarter of the fee. (Given that examination boards are not-for-profit institutions, registered as charities, who are legally obliged to ensure that their finances don't go into the red, I can't see that happening).