difficult one to answer without seeing the boiler or knowing what make and model it is. some combi boilers (like ferrolis) dont have diverter valves at all and others have electric diverter valves.
it sounds unlikely that the diverter valve wont switch over when running the water for a short time but does if you run the water for a longer time, makes no sense at all.
its more likely that you have an intermittant fault, possibly caused by a damaged diaphragm inside the valve or the valve being sludged up internally.
intermittant faults are very dfficult to diagnose as you can guarantee it wont do it when the heating engineer is there, far easier to find a fault when a part is definately broken.
if you wanted a new diverter valve fitted to see if that solves the problem your looking at about 125 to 150 pounds depending on model.