Caution
If the radiators heat but the cylinder does not, the boiler is working and something is not sending its heat to the cylinder: the hot water timer, the cylinder thermostat or the valve.
Low risk
Hot water has to push the cold water already in the pipe out first. The longer or wider the pipe from the boiler to the tap, the longer the wait.
Hot water that arrives but is only warm, or runs out quickly, usually means the temperature is set too low, the coil is furred or the store is too small for the use.
Describe the job and get a fixed price from a local heating engineer.