Sounds very much liker your secondary heat exchanger (which does the hot water) is partly blocked. If the boiler is responding to demand (by firing up) and then keeps going all the time the HW runs with no error indication, then it's probably not a control problem. Fortunately, taking off the HX and checking for blockage is easy. But it might not be the cause....
1. Switch the CH and HW off for a while so thet the pipes cool down.
2. Assuming you have the M&I manual for the boiler, follow the instructions for removing the outer casing.
3. With the CH OFF, turn on a HW tap and check whether the pipe leading to the RADIATORS from underneath the boiler (22mm, third from the right) is getting hot. If so, there's a problem with the 3-port valve inside the boiler.
4. Locate the secondary heat exchanger and the internal pipes leading to it. with the boiler running on HW, these should be HOT! (take care not to burn yourself and also beware of electrical hazards). If the supply pipes to the HX are HOT and the delivery pipe to the taps (first on the left below boiler) is cool, then the HX is the problem.
Procedure for removing it is on page 38 of the M&I manual. If it's blocked on the primary side, physical flushing may work. If it's limescaled on the secondary side, you could try descaling it, otherwise replace altogether.