I have lived with both electric instant showers and showers connected to a gas boiler. And run costs must be down to time in the shower.
With a Main 7 gas DHW heater there was a minimum water through the unit to keep it turned on, so the shower head was larger than with an electric, the boiler would modulate a little, but likely running at around the 15 kW minimum output, the boiler was close to the shower, but the pipe work was at least 22 mm, so a far bit of water needed running off before one stepped into the shower, so the big question is did I use 4 times the amount of water with a gas powered shower compared to electric? I would say very close, but the gas shower with the extra water was a better shower.
Mothers house, the gas powered shower was originally taken from the water storage tank, it was a power shower, however since tank upstairs and shower down stairs, did not need the power turning on, there was no heating of the heat exchanger so water arrived at the shower faster, also pipe work was ½" plastic from the tank so less water to heat up, and no need to have large shower head, so would think no question the gas powered shower was cheaper, can't count the losses from water tank, as would still be heated even is shower not used.
However the cold water tank started to leak, so dad had new central heating fitted with a combi boiler. I had to get the firm back to remove the power shower as illegal now connected direct to incoming supply, as with my shower the rose had to also be changed to a larger one so the boiler would not cut out, and the distance boiler to shower was double the distance of hot water tank to shower, so more pipework to heat up.
However the main problem was the boiler had a small reservoir of hot water to reduce the time it takes to get hot water to the taps, but this was not enough to run a shower, so one had to turn on shower, wait for it to get hot, cold, then hot again before stepping in it. So the water wasted was far more than water used. So once the combi boiler was fitted, likely the gas shower was more expensive than electric.
However with loads of water the experience was better, and after a shower the wet room was really wet, so the problem was to dry the wet room, as transferring from wheel chair to loo one wanted a dry floor, so since needed drying winter and summer, it had electric under floor heating, so by time cost of running the heating is included, having a large gas shower rather than a dribble of an electric shower, clearly cost more.
But the three accounts show how it is not as simple as giving a yes/no answer, it depends on the pipe work used, and how the boiler works, there was an Eco setting on the gas boiler, which if used stopped the cold, hot, cold, hot running of the shower, however swings and roundabouts, as then more water used at the kitchen sink.
I have an electric shower in this house, no idea of size, fitted when I moved in, I am sure using oil would be cheaper, but the cost of installing the pipework with the lifting of the floor etc, likely would not see much change from £1000, so simply not worth doing.
And in the main that is the point, to change from electric to gas, or gas to electric costs money, when starting with a bare room with no shower fitted, cost of running cable or pipes will cancel each other out, so little in the installation cost, likely gas/oil does cost a little more, but the gas/oil or stored water shower is so much better than electric then I would go for gas or stored water every time, but on costs alone not sure, for a proper wet room shower
you need stored water, be it electric, gas, or oil, boilers can't deliver enough hot water to supply a shower like that, it has to be stored water, but for me, that is a pipe dream.
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