I cracked open my cheapo Atten soldering station to see what that was like. It has a 1A fuse on the mains input. The transformer has a fuse symbol on it's label too, implying that is also fused! The label rates it at 5A. And to top it off, the (captive, so you can't misplace it) mains plug is also fused at 13A.
If the Weller did originally come with a fused mains cable/plug, they should have made it captive so it can't easily be lost and operated with a non-fused input. Plus then someone in Dave's situation would not have plugged it in to 240V because it has the wrong plug ("What's this bloody yank thing doing on here? Ah, it's 110V...").