It's a problem that crops up every 18 months or so. The washbasin is one of those which has a pop-up plug, actuated by a rod running through the side of the mixer tap. This particular version is a PITA, as the plug is captive to the lower rod, which means it has to be dismantled to use a plunger effectively. I do have a more recent waste fitting with a pop-up plug that sits on top of the actuator rod, and which should just be lifted clear out to plunge it (provided it works, and doesn't fall off the rod or disappear down into the trap). I've just received a couple of plastic suction pads, of the sort used to change recessed light bulbs, so I can keep a grip on the plug(s) easily.
I've used a chemical drain unblocker, which improves things, but doesn't entirely clear it.
These captive plugs pop up to leave a gap about 1/8" between the plug and the waste fitting, through which the water escapes. This isn't really enough gap to allow the water out with any velocity. The newer pop-up may be better, as it should be possible to just lift it out if needed. However, the long-term plan is to go back to tried and tested means, which is to replace the whole waste with one which has an old-fashioned rubber plug on a chain. This will let the water out quicker and faster, and make plunging easy. Like a lot of modern things, the new plugs etc. are great when they work, but a nightmare if anything goes wrong.
