Good idea but not the Punto 60 camshaft - that would be a FIRE engine, and in NZ, hard to get. Punto 90 camshaft - perhaps, but again that would be hard to get independently of the 1.6L engine, which would be a useful upgrade in itself. Then again, the whole Punto 90 is a nice enough car
The 1301cc motor is similar to the old FIAT 128 motor (there are some of these 1116cc/1290cc engines still knocking around in NZ).
Without a motor transplant, you could apply some tricks used for the FIAT 128. They centre around cylinder head modification, a camshaft change, and larger carburettor or twin carburettors. And, you'll want the electronic distributor (as you mentioned) off a later Uno.
The Uno would be off the road for at least a week while you change parts - is that likely to be a problem? (i.e. it's not a weekend job

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If not, you could remove the cylinder head and compare to a 1116cc head, either off an Uno 60 ('86-'90 in NZ) or off a FIAT 128 (body long since rusted-away but engines sometimes survive, check with your local FIAT specialist). I think the '1100' head has smaller combustion chambers than the 1301/1299/1290cc head, which will bump the compression ratio a useful amount. That by itself will improve the engine's efficiency. As the cost may be just about free, it's good value. (I'd do it

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Cylinder head work is an art in itself - at the very least involving smoothing the ports a little, re-grinding or re-cutting the valves if necessary, fitting larger valves if the budget allows, etc.
To get anything from cylinder head work, you need to change the camshaft for a free-er breathing alternative. A 128 Coupe camshaft may provide some improvement, ditto the X1/9 camshaft, but nothing dramatic. For $$$ you can purchase a new performance camshaft, or you can send off a 128/X1/9 camshaft for re-grinding (about $200). I don't think that re-grinding an Uno camshaft is a good idea - the standard profile doesn't have much metal to work with.
Then, having changed the camshaft, you can think about a Weber 34DMTR carburettor off a Lancia Beta, or a special manifold and twin Weber downdraft carbs. A Kiwi, 'Ladaspeed', on this forum, demonstrates the twin carb principle in the X1/9 section - see his thread and perhaps message him about the manifolds he had made.
In my opinion, these sorts of modifications are not very cost-effective in today's age of cheap cars.

You'd be looking at a power increase from 68bhp to perhaps 100bhp, but you seldom get something for nothing and the engine's smoothness at idle, noise, and low-down torque are probably going to change. Costs will be pushing four figures in $NZ. (My 192bhp Alfa 164 cost $1000 to buy and there's similar cars on Trademe.)
Don't forget all the small jobs like setting the valve clearances (special tool and range of shims required), jetting the carburettors (alternative jet sizes required), and items such as air filters for twin carbs - these three items by themselves could set you back $400.
Of course, if you do go ahead, you create something special that will always be "the hot Uno"
-Alex