OK, thanks. A typical hot water cylinder holds about 100 litres of hot water, provided it has been heated up for long enough (Agas are quite low power and may take many hours to heat it fully - how many hours has the Aga been running with no hot water used?)
Is the cylinder, and the pipes to and from the Aga, well-insulated? What sort of insulation? Feel the primary flow and return pipes at the cylinder, and see how hot they are. They shoul be too hot to hold. Then feel the flow and return pipes at the Aga.
Is the primary flow from Aga to Cylinder pumped, or gravity flow? How hot is the pump?
When the water runs cold, feel the cylinder again, top, middle and bottom, and see how hot it is. You may have to cut or move the insulation to do this.
Presumably the HW Cylinder is fed from a cold tank in the loft? Does the cold tap water come from the same tank, or from the mains? Does the cold tank seem unexpectedly warm?l
Look at all your mixer taps, on sink, shower, bath, basin, bidet etc. Run each hot tap until it comes cold. Then after 5 minutes feel the hot pipe that goes to the tap, and see if it is still warm, or if any of them have gone daed cold (thsi can happen due to backflow, it is espoecially common with ceramic taps)
When was the hot water supply last working properly? What has changed since then? Does the Aga also run a towel rail or radiators? How hot are they? Does the HWCylinder aslo have an immersion heater? Does that give better results?
Does your power shower have a pump, and has it been running? It should have isolation valves, if you turn these off, does the situation imporove? Or does it run at mains pressure?