The UK Air Passenger Duty is indeed refundable if you miss a flight. However for a journey on a UK domestic flight the Duty is £13, (but much much more for long-haul flights).
This is the largest single tax, there will be other small items that are refundable too. BA charge an administration fee of £15, so you are unlikely to come away with more than pennies, it may well cost more to apply than the refund you receive. Very different if you're flying long haul, or in Club/First classes (which don't exist on domestic flights).
Obviously if the reason you weren't on the flight was BA's fault you wouldn't pay the admin charge, but here if it's anyone's fault Royal Brunei are to blame.