Have had several Uno's, what year, carb implies conventional
distributor and ignition coil. Adjusting the timing, and it starts,
may be a red herring. Timing would have to be quite a way out
to stop it starting, I assume the first thing you did was to mark
the postion of everything, so you can see if they have shifted.

Vacuum advance unit ok. No punctured diaphragm? You advance
timing car starts, vacuum advance also advances timing.

You may have muck in the carb, blocks a jet, engine stops,
wait a while, or have air cleaner off - remove jets - blow
through, and all comes good. If it's a lump of muck it
is held blocking the jet by suction, engine stops, suction
disappears, blockage frees, for a short time.

Engine won't start, if your timing light is a simple series
flash tube, wire in series with a plug and see if it flashes.
Or get one of the neon tube testers that go in series with a plug.

Fuel pump pumping ok? Coil ok? Hall sensor, electronic
unit, driving coil ok?

Quite often you can have a heat related problem which
comes good when everything is back to ambient. Some
ign. coils fail when hot. Just a few thoughts.