I don't know if my local bike shop were winding me up. When I took the bike in, as I handed it over I noticed a massive spider emerge from near the throttle. The guy saw it and somehow made it vanish. But it took them less than 10 seconds to determine the brake shoes were worn, they only squeezed the levers! They told me to try the Electric Transport Shop, which I did today. And my biggest fear was confirmed, they are too busy to work on it for the next few weeks.
If I knew what I was doing, I would be taking those brakes apart right now to measure for new brake shoes. But I don't believe they are actually worn.
The rust at the rear brake was nothing. It is just on the nut that secures the cable to the drum brake (it actually says 'Servo brake 98' on the side). There's no way that the brake shoes are worn because they do stop the wheel rapidly, too well in fact. it just doesn't work when I pull the brake lever.
As for the front brake, I turned the nut near the wheel and it stopped working completely. I then gave it another turn and it suddenly gave a reasonable amount of braking force. Another theory I have is that the barrel adjusters have become damaged (probably by me using excessive force to adjust) and no longer tension the cable at all.
I actually know nothing about bicycles, this is the first one I've actually owned, though I did use abandoned bikes for a short while. Those all had faults with brakes/chains and tyres and they were eventually dumped again.