Quarter turn bathroom tap: Valve removal
Solution!

I bought a box spanner and removed the entire unit (including the silver connecting hoses) and then went to a few local plumbing shops with the whole tap. 

It seems to be a rare type of unit for London.

Step 1: Turn the water off at the stopcock.

Step 2: Pull (Yes pull) the handle out. The handle had two rubber rings holding it into the concealment cylinder (The cylinder between the handle and the main shaft).

Step 3: Unscrew the concealment cylinder (similar to video 2 in this clip: https://www.youtube.com/watch?v=CQc5WWbKmnI).

Step 4: Unscrew the screw holding a plastic connector for the tap out of the cartridge.

Step 5: Remove the cartridge and replace.

Step 6: Put things back together :)

Good luck!