The slot that the stud locks in to is part of the backing plate for the disc and it had bent outward and rounded off over time, no longer giving enough metal for the stud to hold on to. I managed to build the slot back up using a cole-chisel and there is just about enough metal left to hold the stud firmly. If it happens again in the near future then a new backing plate may be required, but this is a hub off job and a whole load more work, so lets hope it doesn't.
Annoyingly, I had to crack open a new £13 handbrake spring assembly kit just to use the one sprung retaining stud, guess I have some spares at least. With the left side now working correctly, I gave both sides of the handbrake a quick tighten up using the rotating adjuster inside the drum.