This is plasterboard or something with a void about an inch back
This is the wall plug
The hole here is actually so big that the wall plug would fall through into the void if I put it back
I can't move the radiator up or down or left or right because it is fixed and even if I knew how to disconnect it I wouldn't want to spill black water everywhere and ruin the carpet, so I am thinking I should use the same holes but get a bigger plug
What about the plastic ones for plasterboard that sort of butterfly out when you fix them in?
How much room do you have between the wall and the radiator? If you can't move this to attach directly to a stud, I would mount something on the wall horizontally that does attach to the studs, and then attach the radiator to the horizontal. Something thin like a 1x4, painted to blend into the wallpaper so it's not very noticeable.
Nice OK, I will see