It seems to have worked!
The coil you mentioned in the outside transmitter was only soldered at one end, thus I presumed it must definitely be the antenna, so I jumped in at the deep end! I didn't replace it with a 16-17cm wire, but just pulled it out straight. It measured over 20cm (should I cut it back to 16.5cm?). I then cut a slit in the case to exit the antenna and screwed back on the case.
As the antenna was anything but rigid, I slipped a stiffish tube over it, which I then taped to the side of the unit.
The whole I returned to my weather station box (cheap version!) made up of 10 flowerpot saucers painted white (the white paint must be renewed soon!!!). All the 10 saucers ar fitted vertically over each other, with spacers between.
It's now been running for a number of hours, and the outside values (T and RH) still show no flashing of last values measured. The radio connection is still being upheld.
Many thanks again.