Further experience to add to my original post:
Well, I thought I had cured my drip-drip-drip problem but with the flash-flood that we had last night and this morning the problem has resurfaced.
So, donning waterproof trousers and anorak I braved the weather to see the problem in action.
The V-slot I cut in the overlapped downpipe wasn't large enough, so I used a junior hacksaw and widened it.
Whilst I am up there I spotted that rain was dripping off the flat roof felt straight down the pipe - perhaps this was partly to blame. As an explanation to the above, a flat roof has a long horizontal length of U-shaped guttering to catch the drips off the felt and channel it to one end where there is a downpipe. This is fine for the majority of the water since any drips are two or three inches from the felt to the gutter (and often into a tiny stream of rainwater which must further cushion the splash). But the three inches of flat roof immediately above the downpipe suffers the predicament that any rain drips coming off that bit of flat roof will fall all the way down the downpipe - drip, drip, drip... aaaaargh.
All it takes is a less-than-perfect flat roof with a slight hollow in the bit that is near the downpipe and more water will escape over the edge of the roof there.
I'm wondering to the merit of always fitting one of those S-bends (zat called a "goose neck"?) near the top of the gutter so that vertical-falling drips must hit plastic after about six or more inches therefore reducing the dripping.
I don't have the kit just now, and such a change would either involve complicated ladder arrangements or hanging over the edge of the roof, so I have implemented another idea.
I found a length of guttering about 6 inches long and I have cut about one inch off the long edge. If anyone repeats this, cut it carefully and be prepared to cut more off it to suit your own circumstances.
The cut bit of guttering is now installed upside-down in the existing guttering above the downpipe. Any drips will hit this additional piece and deflect to the side before being channeled away down the downpipe's sides.
All for a quiet night's sleep...
Mungo
P.S. No doubt that by posting this follow-up I will tempt fate yet again... :-)