Firstly, thanks for all the replies above, I really appreciate that. I've been offline for a few days so excuse the delay in getting back.
dub-nerd - the leaks are always in the radiator panels themselves. The discarded radiators look like a bubble was created in the metal, and that bubble burst.
Shane007 - I think I only have a large tank in the attic (no little one). There's a smaller tank (cylinder) in the hot press though.
Just to update, the plumber has replaced one of the radiators last Thursday (and switched the other one off). He put the oxipic into the tank that is in the hot press. I suppose there's no way of knowing whether that worked or not. We'll just have to wait and see if any more radiators leak.
Later our hot water started to take on a reddish-pink colour. We told the plumber and he was amazed as he says the hot water system and heating system should be separate. He now thinks there must be a leak in the cylinder, so we probably need a new one of them.
He said that hopefully the reddish/pink colour would wash through in a few days. It had washed through by last night.