Can you run four electric showers from a single 15mm cold water supply?

I would doubt it, that's just a bit over 1/2" and suitable for 1 feed line. While high pressure may help, you would be low on volume.

You could - if you used only one shower at the time. The minute somebody turns on water in another shower, your volume will drop down considerably. There is simply not enough water in the pipe to feed such demand. As suggested above - 1" would be optimal size of water pipe feeding 4 showers.

Milo