You most likely have a regular conventional boiler, with a cold water cistern (bigger one) and F&E cistern (smaller one), likely non condensing. It's got an indirect hot water cylinder. We can see that the whole heating system is gravity fed/vented/unsealed. The red thing is a pump, so it's either fully pumped or semi gravity (with heating pumped, but the water coil left to gravity and not pumped). It is unlikely to be full gravity (the original unpumped system with thick vertical pipes that encourage convectional circulation). The hot water system is also gravity fed – a centrifugal twin impeller pump can be used to boost flow rate for shower, toilet and bathroom hot water from the cold water cistern through the cylinder and cold water from the cold water cistern; single impeller pump does either hot or cold, and usually goes between the mixer and the shower head and not before the mixer, so that it does both hot and cold. A pump is not absolutely necessary, because water always flows towards the outlet, unlike in a heating system where there is no outlet or pressure drop.
If it is not fully pumped then it is a semi gravity which usually uses a C thermostat plan. Find the motorised valves and their locations and appearances, this will tell you whether it's a C, Y, W or S plan.
The first upgrade would be to get sealed system equipment or a system boiler (which integrates the equipment into the boiler). This equipment is a filling loop to connect mains to the central heating to allow for manual repressurisation (if there is a leak) to replace the automatic F&E tank repressurisation supply, an expansion vessel to replace the expansion vent in the F&E tank, and a pressure gauge. A system boiler has this equipment inside the boiler, and also integrates a pump for the heating circuit, which is is now W, S or Y plan; W uses a diverter valve meaning only the radiators or the cylinder can be heated, but not both. Y plan uses a mid position valve allowing both radiators and cylinder to be heated at the same time, but there are only 3 positions so there's no real temperature control at the valve, instead it's time based (the valve is open more or less frequency) and not like a mixer valve. S plan has 2 separate pipes that branch off the CH flow pipe, and there are 2 separate motorised valves which are also either open or closed.
The water coil and heating are now fully pumped and unvented/pressurised/sealed, but the hot water cylinder is still gravity fed via the cold water cistern. To do away with this, you need an unvented cylinder that can withstand mains pressure, often made of steel, and the hot water circuit needs to be pressurised by connecting the mains directly to the cylinder inlet and fitting an expansion vessel in between, as well as another pressure gauge, backflow preventer and pressure reducing valve. The cylinder is now direct fed, hence pressurised, and doesn't need the cold water cistern or vent. This is now a system boiler with an unvented water cylinder. The cylinder could be upgraded to have an immersion heater as the main heater which heats the water without gas or the boiler workings. Some cylinders, particularly vented ones, have a backup immersion heater for the summer.
In cylinder systems, the hot water is not on demand. Instead, the thermostat on the cylinder sends a signal to the programmer panel when the water in the cylinder goes below a certain temperature, and if the hot water is on, it sends a signal to the motorised valve and then the boiler.
You could instead upgrade one step further. Get a combi boiler and get rid of the cold water cylinder. The combi boiler contains the expansion vessel and the heating pump like the system boiler, but the secondary exchanger is a plate that exchanges heat between the CH and hot water pipes. The programmer panel is usually incorporated into the boiler and the thermostat connects directly. There are usually 5 pipes. Gas in the middle. Central heating flow and return are the 2 outer pipes and then cold mains on the inner right and hot water on the inner left. The hot water circuit is pressurised because it comes from the mains, and the central heating is pressurised using a filling loop from the main to the central heating return. There is a further copper PRV pipe, and a plastic condensate pipe, and a flue pipe. The burner gets activated by the thermostat for central heating demand, and the combi boiler probably contains a dynamic pressure flow detector to detect hot water demand, and the burner starts up, and a motorised valve inside the boiler turns to direct the CH flow to hot water (or mid position between hot water and radiators), which passes the heated CH water through the plate heat exchanger which conductively transfers heat to the hot water pipe. There's no expansion vessel or pressure gauge for the hot water because it's just hot mains in a pipe without a cylinder, also its only being heated when an outlet is open, meaning there's not enough volume of water to expand and cause problems. I know potterton gold combi valve either has hot water or hot+CH mid position. This way, the plate heat exchange path is always open, and the the condensing gasses on the secondary heat are always inputting useful heat.
In your current set up, the shower needs a shower pump, or instead you just need to upgrade to a system boiler with an unvented cylinder, or a combi boiler, to get full pressure in the shower, which typically provides a higher flow rate than a pump does up that elevation.