As you have a cylinder and a cold water tank, you can have a pumped shower if you want. A few showers are available with their own pump inside the casing on the wall. You may need to fit a new, larger cold water tank, with a close-fitting plastic lid and an insulating jacket. If the old one is small it might run dry when pumped, which is tiresome. The lid will keep wildlife out. You can get good results with a modern tank and cylinder.
You can heat a cylinder with a backboiler, or a gas boiler, or a multifuel stove, or an immersion heater, or any combination of them. The immersion heater will cost the most to run. Modern practice is to have larger, well-insulated cylinders as small ones will hardly do one bath or big shower. It does not matter if they are slow to heat if they hold plenty of hot water.
Only pump the shower, as it will be annoyingly noisy. You may need to run new pipes. If you use chromed ones, coming down from the ceiling to the mixer, they are not unsightly.
Have a look at the incoming pipes. If lead or iron they may be small and leaky. If lead, ask your water co to test the drinking water for lead content. There might be a lead replacement subsidy.
Run the bath hot tap into a bucket, time it to full, calculate how many litres per minute it delivers.
Then do the same with the shower, then take the head off the hose and see how much the hose can deliver.
Then do it at the COLD tap at the kitchen sink.
What are the numbers?
This will help determine if your house can support a combi or an unvented cylinder.