The light on the pull cord tells you the power has got that far - the light is wired to the load side of the switch so it should also mean the power is reaching the shower unit. If the light isn't on it means either the lamp has died (does happen, it's only a light bulb after all) or that you've got a problem in the pull witch.
Often found in shower pulls is one of the wires not tightened enough in its terminal and over time arcing causes it to degrade and lose contact - it will often cause the RCD to go as well, but not always. If your cord light is off the first place to look is in the switch. Make sure the circuit breaker at the consumer unit is OFF, open the pull witch and look for any signs of burning or crispy insulation. If it's there you'll need to trim back the cable to soft, pliable insulation, and probably replace the switch as the arcing will have damaged the terminals.
You might have got away with just a loose terminal, in which case tighten them all up and try again.
If the cord light is on, check that power is reaching the shower unit (lights should be on, or open it and check with a meter). You could also have a loose connection at the shower unit end, so have a look at those terminals.
If power is reaching the shower but it's not working it'll be some component part of the shower; the manual will give a few ideas for problem solving. Clogged filters can be an issue, thernostat failures another popular one. Or as you suggest, the pump motor ....
PJ