Shower pump humming when taps opened, but no flow
Turns out the whole air/negative head/priming thing was a complete red herring. I found the manufacturer's FAQ and in it they state that if the pump is humming then it is likely physically not turning.

So I disassembled the pump and on one side, found that a small retaining spring pin had come loose and had jammed into the impeller, breaking a few of the teeth. Once the pin was put back in its proper place and the pump reassembled, it works again.