Hi Stijn, welcome to the forum
You would have some difficulty fitting other Uno gearboxes to the Turbo because the driveshafts are different. Specifically the Turbo has inner CV joints rather than the 'Tripode' universals that live in the differentials of the lesser Uno models.
Ignoring the driveshaft difference, the closest is the Mk1 Uno 70 about 1988, which came with a similar gearbox to the original Mk1 Turbo - a single gearchange rod going into a 'bellows' at the bottom of the gearbox. Most other Uno 60s and 70s have a different style of gearbox with a gearchange linkage that has a vertical bar at the back of the gearbox and a rectangular 'tin' cover on top (this gearbox is similar to the one used in Uno 45 models). Fitting this to the Turbo would be a waste of time as it's probably not strong enough.
The later Mk1 Turbos (like mine), seemingly quite a number of them in 1988, came with a Tipo-style gearbox having two balljointed rods on top and the gearchange linkage inside the car rather than underneath. I believe this later style of gearbox is much stronger and you should be able to find a working replacement out of a Tipo. However, there are more driveshaft differences - the 1372cc Tipo comes with Tripode joints (again), not sure about the 1580cc models. I expect that internal components of the gearbox could be exchanged.
Sorry this possibly isn't the news you were wanting, but bear in mind that you may be able to use FIAT Strada parts to repair an early Uno Turbo gearbox?
As for suspension, I've put some effort into lowering and stiffening the suspension of my Uno Turbo. You cannot cut the springs, due to their conical shape, so you have to fit replacements. Quite easy to do, particularly when the original shock absorbers are likely to be due for replacement by now. I feel that the next biggest weakness is the driver... !
Cheers,
-Alex