An immersion heater warms water at the rate of about 1 litre per minute. Unless you have an unvented (white) one, the pressure is quite low, but you can get a shower pump to use with any mixer, or some showers include their own pump (tends to be more expensive). With a pump, the cold water has to be pumped from your loft tank, not the watermain.
You can get a good forceful shower using a pump, and because the water is heated in advance, you might get, say, ten minutes at ten litres per minute before it runs out. Longer if it is mixed e.g. 50:50 hot and cold.
This would be the most economical way of changing from bath to shower.
When you have several people wanting showers, there can be an annoying delay before the cylinder re-heats.
(modern gas boilers have about ten times the power of immersion heaters, so they reheat the cylinder faster)
Because unvented (white) cylinders can deliver hot water so fast, people usually have rather big ones. 250litres is quite common. This is rather heavy and they are often installed on the ground floor. If you are considering one, you may need your incoming water pipe upgraded to deliver water fast enough. This is a very good solution, but probably the most expensive, especially as you do not have a gas boiler.
Because electric showers are so weedy in water delivery, they are often fitted in a "second shower room" to avoid queues for the bathroom, when you might accept the weak flow.