The cylinder is extremely old. The white jacket predates the more recent red one. Red jackets have been around for about 40 years.
Start thinking about replacing it when it starts to leak or some other problem occurs. You can get a modern equivalent which will heat up faster, be much better insulated, and I suggest considerably bigger. The one you have is about 100 litres which is adequate for one bath. You do not have to change to an unvented cylinder, and you would probably need to upgrade your pipes if you did.
I can't see a thermostat or thermostatic valve on your cylinder. This is not unusual on such an old one, but is uneconomical on gas. Probably it heats up all the time the boiler is running, , even if it is already hot enough. This would lead to hot pipes under the floor.
The immersion heater is also very old and is capable of going faulty and overheating the cylinder which can be very dangerous (modern ones have an overheat cutout as well as a thermostat).
Use a thermometer to test hot tapwater temperature.
If you can find a trustworthy independent local heating engineer you could have a new cylinder, fitted with thermostat, converted to fully pumped, with motorised valve and insulated pipes, which would give you better economy and more hot water.
If you want to spend more you could get a new boiler and an unvented cylinder and new water supply pipes, which will be much better, especially if you like multiple baths and powerful showers.
If your house is small with only one bathroom and only one or two occupants, you could have a combi
I would avoid doing anything to your old cylinder because it might leak.