DIY

If I had a volunteer to unbung the sink, who could and would, with a smile, do the job, without guess work, without making a mess and a terrible smell and without involving my services, I would have joyfully accepted the offer.
Short version.
The way things are. 
I waited until I was alone in the house, dismantled everything, cleaned it all in hot water, soap and soda and put it all together again.  
Easy. 
Just time, patience and mess.
Unfortunately the sink was still blocked, so I realised that there was another, probably hard, fat blockage  in tube that enters the wall. 
Rather than dismantling the whole thing again, I poured baking powder, vinegar and boiling water down the sink, which actually did the job.
I spent yesterday and the day explaining a lesson in italian to foreign english speakers.
Interesting but very stressful.