You just couldn't make it up Michael ;o) Strange set of circumstances.
I guess that, first of all, the mains pressure is very poor. When one valve is open, everything else is starved............ maybe
Apart from that, I would replace the valve coming in to the cistern. It sounds as though it's bouncing, and can't make its mind up whether it's shut off or not (that's possibly a float problem.) The float may be fouling something. In which case, as the water level goes up and down, the float jiggles about and lets in more water. The shower and basin, I guess, are simply causes the water pressure to vary....... thus "confusing" the cistern valve.
This is all speculation, but there is no other interconnection between the various appliances.
If the float is sticking somewhere, it's an easy job to adjust it, or replace.