Thank you! I already have stripped the Japy, but cannot see anything wrong. Cannot see too much though, it's really small and packed inside. I am thinkng of taking off the entire carriage with its rails to be able to see what happens underneath ... things can't get worse anyway. I will probably strip the Tower too, first, in order to be able to compare better. On top of everything, this was my first machine with a french keyboard. Sigh.

Good and bad news: By chance, I got another Japy Personelle, this one with a German keyboard. That one is fully working, a little dusty, no debris whatever, no grime. Only, sigh again, with a damage on the front frame, from falling, I guess, which led to the space bar just not having enough space to go down.
After meticoulosly comparing the two, I still could not figure out the shift mechanism problem with the first. So I decided to abandon it, at least for the time being and put the body of the first one on to the second one, so that at least this one would be working. And it is. Double easy and fast typer! I also put the back bottom cover from the first on to the second, because it shows less signs of usage - and in doing so I noticed that it is bent. Very slightly, but bent. So I guess this machine has suffered a shock, too, which might have led to the dislocation of the carriage. I  think that this has not happened during transport, as, despite only being in a soft bag half-loosely set in a cardboard box, the machine itself was attached to the bottom plate, and the soft case was padded with newspaper inside. No chance to bounce around. 
In the image, you can see the difference in the position of the carriages. The left one is the frozen one, the right one  is in regular up postion.