I've just read all 12 pages, the biggest clue to date seems to be that the immersion heater is capable of producing hot shower water and the boiler is not. The pump sounds as though it is functioning.
Yet the water from the tap is a decent temperature when the immersion is not used.
As this is a very strange set of circumstances it's making me wonder if the tap hot water is actually coming from the cylinder.
Could it have it's own under-sink heater?!
Or could something very special have happened, such as the boiler being changed to a combi, the tap being supplied by hot water on demand from the boiler, the cylinder coil ended up connected to the heating flow/return, and the shower being the only thing fed from the cylinder. I realise that is exceptionally unlikely, but so far something like that is the only thing that seems to fit all the reported test results.
If the OP can run the basin tap (as suggested before I think) and confirm the header tank for the cylinder starts to fill up?
Have we had a picture of the boiler yet?