Taps:
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Sounds (from a logical view, not an expert on Combi boilers) that whatever detects that water is flowing and therefore needs heating (pressure valve?) is sticking a bit.
Open tap - no action. Open tap, close (pressure raised above norm briefly) open again (larger pressure drop) and the bolier kicks in. Effectively you are 'kicking' the pressure sensor. [Could low mains pressure make this happen also?? i.e. not enough difference between taps closed and open to trigger the pressure sensor.]
Once the boiler is going, presumably (?) it is turned off once max temperature is reached, so hot water continues to flow.
So your problem is getting the boiler to turn on not turn off.
This is supported by the fact that once the boiler is running (for the central heating) you get hot water straight away. [Although presumably this would work even if the pressure sensor was totally non-functional?]
Central Heating:
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O.K. at first, but now whatever senses that the water is under temperature in the central heating circuit (or just the boiler) is not functioning (boiler thermostat?). So although water is being pumped round the boiler is not kicking in because it cannot detect that the water temperature is low (switch on).
However when you start the boiler (hot water tap pressure switch) the boiler starts, and then heats the water in the radiator circuit also. This suggests that the pump is running but the 'on' thermostat is not working.
One thing is not clear from your account.
Does the central heating only work whilst the hot tap is on? If so, the thermostat in the central heating circuit is completely thrashed.
Does the central heating warm up fully once a hot tap has been run for a little? If so it is only the 'on' action of the thermostat which is not working. [And you are going to spend a lot of time turning hot taps on for a bit to keep the house warm]
As I stated at the start, this is a (hopefully) logical analysis of the symptoms - I don't know the bits of a Combi but I am assuming:
Hot water boiler on - pressure sensor Hot water boiler off - h/w thermostat CH boiler on - c/h thermostat CH boiler off - c/h thermostat
This assumes that the valves and pump are working correctly and the timer/programmer is supplying the correct voltages.
Reading through this there are inconsistencies (you talk about the boiler 'cycling' so at some stage the 'on' and 'off' functions of the thermostat are working) but hey - logic is at best a weak and damaged tool :-)
There are also more complex interactions e.g. the hot water flow to the taps is getting too hot (so turn the boiler off) but the CH water is cool (so turn the boiler on). I presume the HW would over-ride the CH in the interest of safety but this shows that a simple analysis could be a load of tosh :-)) Also that you could just have a thrashed logic circuit in the controller.
Summary:
Sound like you have more than one problem You may have problems with your pressure sensor and your boiler thermostat. Or something else entirely.
Cheers Dave R
P.S. we seem to have a rash of 'my combi system doesn't work after the summer layoff' threads at the moment. So:
(1) As I was advised with car aircon over winter "just because you don't need it don't ignore it; switch it on at least once a month to keep it happy" so I would suggest that the CH on combi systems should perhaps be turned on at least once a month over the summer just to check it is still working and exercise the controls. At least you will know before winter if you have problems.
(2) My old conventional boiler is still running well, and we have been in the house nearly 20 years. Lord only knows how old it is, but it passed the last Corgi inspection with flying colours. It is less efficient than a modern combi but all the bits are obvious and easy to replace. Projected savings with a new combi are all very well, but if bits keep dropping off then the savings may not materialise. My mum-in-law is on her second Combi in 10 years (mind you the first one was cheap crap). I am drawn to a comparison with cars - my old Moggie Minor was slow and inefficient, but I could fix everything on it. My more recent Volvo Turbo Estate (now quite elderly) has so much electronic gadgetry under the bonnet and fancy doodads and chips and analysis sockets that it frightens me every time I open the bonnet. Faster, more powerful, etc. but a bugger if it goes wrong. So is simplicity best at times?