It's the height difference between the water level in the COLD water storage tank in your roof to the shower head which affects your pressure and therefore your flow rate.
When you have the shower head down in the bath it's probably about 2 metres giving you around 0.2 - 0.25bar.
When you raise the shower head up to the level of the bracket you reduce the static water head and therefore the pressure drops.
That's why it's dribbling.
The cold / hot differential is probably because you don't have the hot / cold feeds properly balanced.
Incidentally, the I'm not clear if it's a thermostatic mixer or a bath shower mixer. Could you clarify? If its a BSM it will WILL reduce the water flow because of the divertor but that's not your primary problem.
Has the mixer got separate feeds from the CWS in the roof and the HW cylinder. That is to say separate from every other tap? If not it should have. That is easly accomplished for the cold feed but will require a Surrey, Deleted or Warix flange to be fitted to the HW cylinder. Both these feeds should then go to a double ended pump to raise the pressure to an acceptable showering level when the shower head is in the wall bracket.
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