Possibly. But damaged pumps generally send less fuel to the engine.
I wonder where the fuel filter is on the carb model and whether it's on the see-through sort? Can it be seen to fill up and does it empty? A test to check it's working fine would be to disconnect it and put a hosed end into a bottle to catch the fuel. You should get a decent stream.
With the floats, can the float level be set? On the Strombergs, you can set the height at which they cut the fuel supply by bending a small metal tab. New floats on those carbs need setting using a ruler and 90º angle to read off the highest point on the float.
As another thought, our cars are old, so crud could have been sucked into the carb from the tank, although it would have to get past the filter. It's possible something could float around, block the jet and drop out once it stalls. Thus fuel starvation could be the case and not enough fuel is getting to the carb after cranking. I would surmise that the pump runs on cranking, to fill up the float bowl and get the car started but after that can't keep it running.
When you say return line on the pump, does that mean there are three hoses coming out of it? One from tank, one to tank and one to carb?*
Or, if might not be the carb at all. For example, whilst the battery might have enough oomph to start the car, if it's faulty or the alternator's not doing it's thing, it might not be generating enough electricity to keep the engine running (how fast is the starter motor? is it slower than it used to be?). Alternatively, it could be a vacuum leak at the manifold, which would cause lean running and thus cause it to cut out.
Sorry if this sounds like clutching at straws. How Unos work is new to me at the moment!
*Been reading up on the layout of the carb Uno and it does! Wow, that's interesting!
