What a coincidence, my first experience using an electric shower was yesterday, here in Peterborogh where we're staying. I didn't even know such a thing existed.

It certainly sounds like a problem with current draw, doesn't it? Or perhaps a problem with the wiring. If the correct fuse is on the circuit with the power supply to the shower, then there might not be an actual wiring short (the fuse should blow if that's the case). Maybe there's something wrong in the electric unit itself, such as a failing heating element, that's causing too much current draw?

Or maybe it's simple and it's just the switch attached to the pull cord that's shorting out.