Great common Uno question that one!
Have you tried a few Advanced Searches in the Uno section?
I think the most common explanations are: worn rear brake shoes, shoes with faulty self-adjusters, seized curved levers (the parts that go through the brake backplate), seized equaliser (the pulley bracket that swings near the exhaust pipe), seized cable.
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First off, make sure the lever is solidly-mounted. It should not wiggle or bend side-to-side. The lever only has two bolts. You're right-hand-drive in South Africa, as we are, so you'll find that the bolts are in the wrong placer (the front one is on the driver's side when it should be on the other side, to oppose the pull of the driver). The captive nut is fastened to sheetmetal that sits on top of a reinforcing structure - so it just tears off. Of course, there may have been some subtle change to Uno production for ZA, so if the lever feels OK, move on...
Otherwise, to get to the lever mountings, you have to either remove seats and carpet, or cut the carpet between the seats for access
eek
You can either weld the nuts in place, or you can weld an extra tab to the lever bracket (for an extra bolt), OR you can just replace the bolts with longer bolts, large washers under the reinforcing structure, and Nyloc nuts underneath. This requires no welding, though you will have to modify the plastic cover.
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For the brake shoes - take off the rear drums and check the thickness remaining. There should be at least a couple of millimetres friction material thickness. But even if the friction material is perfect, it counts for nothing if the self-adjusters have worked loose. The self-adjusters are the large spring-loaded washers that are part of the shoe itself and these form the main mounting of the shoe. The shoe should be kept close to the drum at all times, so that the handbrake mechanism works effectively and doesn't lose movement just bringing the shoes out to the drum. Check the self adjusters by prying the shoe away from the adjuster (and tapping towards it) - the shoe should be difficult to move relative to the adjuster. If in doubt, fit new shoes.
For the curved levers - while changing the shoes, you will see how simple the handbrake mechanism is. Very easy to de-rust and lubricate the levers with copper grease (high-melting-point grease for brake mechanisms).
The cables under the car deserve a spray with penetrating oil (CRC, WD-40, etc.) and get someone to work the lever so you can see that everything moves properly (pulley and cables).
Don't be tempted to over-tighten the lever. The tighter it is, the less leverage you get. Five clicks, or even six, is OK. It's quite normal for the Uno handbrake to have no effect until the last click or so. The adjustment is therefore critical so that the final click is just 'so'!
The handbrake is only ever as good as the condition of the rear shoes and drums, so give those priority - and attend to any fluid leaks from the cylinders (there should be no fluid under the end dust caps - if there is, replace the cylinders and the contaminated shoes...)
That should keep you busy for a while
-Alex
