I know the line between “technology demonstrator”, “prototype”, and “production model” is fairly subjective one, it from what I have read, all three “employed” devices, Gadget, Little Boy, Fat Man, seem far from what we would think of as “production types” today. Gadget was especially a lab model, but even LB and FM needed to be assembled in situ, by physicists like Luis Alvarez and Norman Ramsey, future Nobel laureates. Far from a weapon which could be sent from the factory to soldiers in the field. And even Gadget could have been jury rigged to be used in a pinch.
It’s like having an F-16 and needed a Lockheed team to assemble a place before every sortie.
Why did they go with that approach and when did weapons become usable without needing scientists to hold the military’s hand.
Did other nuclear powers follow the same path or they waited to have production models?
None of the weapons developing during World War II were production models. The production model of the Fat Man bomb, the Mark III bomb, came in the postwar, and involved standardizing many of the parts and components of the Fat Man. Even then I think one could say that it took a few years until things got truly standardized and not as "one-off."
They went with that approach because they were developing things on the bleeding edge where every example counted. In the postwar, when they started to build up an actual stockpile, they had time to do things the "right" way, once they were sure how they wanted to do it. (The engineers at Sandia were largely responsible for that re-working.) Remember that even with Fat Man they had never "really" tested a production model — they had tested the physics package but not the fully-assembled bomb system. (And there were even several minor changes to the physics package made between Trinity and Fat Man.)
Even into the 1940s it required specialized teams to assemble the weapons. The Mark 3 was more standaridzed than the Fat Man bomb but still pretty complicated. The Mark 4 made huge steps in the direction of standardization and ruggedness. But all things are somewhat relative — it took many years until assembly, arming, etc., became more or less routine (as much as they can be, anyway).
Many nations have fielded "emergency capability" weapons (which is what the US called later "early" versions of weapons, like the TX-14, for example) prior to establishing solid stockpile weapons. Again, the reason to do so is purely because they feel it would be dangerous to wait until getting something more perfect/safe/standard.