Isn't the idea of an electric shower that it saves money by only
heating up what you need? I think the powerful ones are about 10.5kW
... does that sound right?

Yes, the electric shower will be more efficient at point of use, as there
isn't any hot storage for the heat to leak away from. However, electricity
costs many times the cost of gas, so the small leakage you get is
insignificant.

I do and I currently don't (run out that is!). It's enough for 4
people in the course of an hour to all have hot water. My only
experience of a power shower is at the in-laws, where it sucks the
hot tank dry in about 10-15mins ... which is crap!

Well, a cylinder and boiler should be easily capable of beating 10kW
continuously. A modern modulating boiler with reasonable modern cylinder
could run at 20kW or more continuously and never run out.

If you just choose a shower or pump with variable flow and turn it down
rather than having a gusher, you won't run out. Obviously, ensure that the
hot water is turned on at the programmer.

Christian.