Poor hot water flow

I am trying to figure out which part is likely to be faulty and could do with some advice.

Symptoms:

• Hot Water does get delivered but only for a few seconds, less if the flow is high, like the shower.

Tests:

• The heat exchanger is getting warm but goes cool\cold quite quickly
  
• The vent pipe is jacketed but if pull the foam back it isn't warm. Could this mean the tank is too cold?
  
• Running the boost does not appear to help.
  
• I never hear the pump come on when testing but I'm not sure if I should or not.
  
• The header thank is as full as the float-valve will allow. It required a small top up but it was by no means empty.
  
• The case is warm, not hot, to the touch to the left of both immersion elements.
  
• The PCB LED blink pattern indicates the thermostats are both functional.
  

Based on the above can anybody suggest either a fault or further tests I could do? Please bear in mind I'm a consumer not a plumber or electrician.

Thanks.

Ok so further diagnosis seems to suggest that the pump isn't running. I've seen suggestions that I should remove the large brass screw at the front on the pump to check that the impeller blades can spin freely as they may be sticking. Should I expect a lot of water to come out when I do this? I want to be prepared before I start!

Also the Led on the PCB started flashing to indicate a T2 sensor fault briefly before returning to the normal pattern. Could this indicate something else?

I am in the Milton Keynes area if anybody can recommend an experienced Pulsacoil engineer in this area it would be greatly appreciated. Thanks.

Ok I clearly have no idea what's going on and I'm not getting any replies. If I ever figure it out I'll let y'all know in the hope that it saves people grief in the future.

One thing I do know is these Pulsacoil are a huge pain in the arse and I curse the builders for installing one rather than a proper boiler!