Well, apparently the diff out of a Lada would help with my Spider's cruising ability - the Lada diff is 3.9:1, some FIAT diffs are 4.1:1, and mine's probably 4.3:1. I'm pulling over 4100RPM at 100km/h, which is a bit ridiculous for a torquey 2-litre (and noisy...)
I imagine the Lada gearbox must be broadly similar to my Spider's, too, but I think the Lada is only a four-speed. If it is a five speed, that's a tempting swap, as I'm sure nothing could be noisier than my Spider's 'box.
As Steve says, the Lada engine was actually a special design based on an old FIAT engine (FIAT 1500 I think) but with an overhead camshaft. In other words, the Lada engine started off OK by the standards of the time, being an update of an older FIAT design. But it's not the same as any FIAT engine.
To answer your question properly - parts apart from the engine are interchangeable to some extent between the FIAT 124 family (Saloon, Coupe, Spider) and the Lada saloon. But there would be little interchangeability between the Lada and the 128, much less the Lada and the Strada, and then the Uno is even newer than that. So, the answer is basically "No".
The Samara was a similar age to the Uno but the Samara is not an ex-FIAT design - it is a wonderful synergy of Russian and German (Porsche) engineering.
While we're on the topic of formerly-FIAT-based cars, the Seat Ibiza is also from a similar age to the Uno, but again, is not likely to have parts in common (unlike one earlier SEAT which was a 128 Coupe 3P built under license), though you may find the wheels fit. The FSO Polonez saloon likely has some interchangeability with the FIAT 125, and the FSO Niki is just a FIAT 126 made long after it's sell-by date...!
-Alex