Sometimes, a camera just captures a process.
1. Turn the flush buttons piece
2. Remove the buttons
3. Take off the top of the cistern
Now inside the cistern, the float valve is accessible, it just needs to be unbolted. Turn off the water first, then flush to empty the cistern (and put down some old towels because water will spill).
Spotlight with an LED torch. First take the nut off, then the plastic nut.
Out with the old, in with the new...
Tighten the nuts, and all done!