The question title states 'could machine guns be produced in the 14th century' it then goes on to confound itself in the body of text. A modern weapon could not be made by a 14th century nation.
Casting (and thus relative uniformity of production) existed well before the Industrial Revolution though, certainly using different moulds would produce different results, but ammunition could be sorted by hand post-production readily enough, it wouldn't even require any expertise.
One does not need exact measurements nor modern reliability to create a 'machine gun.'
Steel is not required to produce good springs.
Steel is not required to make cartridges.
Steel is not required to produce guns.
Perfectly reliable charges are not required to make good weaponry.
Automation was a known concept, numerous examples exist of it, from clockwork mechanisms to spring powered saws.
From Wikipedia - "Up to the 15th century, clockwork was driven by water, weights, or other roundabout, relatively primitive means, but in 1430 a clock was presented to Philip the Good, Duke of Burgundy, that was driven by a spring."
That is not to say the metal spring was not invented until the 1430
From Wikipedia Torsion springs consisting of twisted ropes or sinew, were used to store potential energy to power several types of ancient weapons; including the Greek ballista and the Roman scorpio and catapults like the onager.
1836 Colt and his revolver. Such a miraculous invention.. with the use of a spring and a ratchet..
The mechanical concepts (and even the parts) required to build a machine gun had already been demonstrated long before the creation of what people recognise as the first automated weaponry.
I think the only part of a machine gun that could not have been readily be designed by a 14th century artificer is ammunition of any reliability, but that needn't kill the project if a part of the automation process expelled failed or partially failed cartridges without relying on the power provided by a spent round.
there is no need for an automatic weapon to fire quickly in order for it to be classed as a machine gun, nor is a rof of 400 or more rounds a minute in any way feasible imo.
A 1300s machine gun would probably make use of a secondary, manually empowered spring prior to contact to act as a guarantor of motion, any failure to fire being catastrophic(which is likely a primary reason why the gatling gun wasn't automatic)
Something like a maxim, but instead of striking a cap a fuse is lit in the rear of the casing with a delay charge, arranged such that perhaps cartridges are lit with a fuse of ~1s.
That is, the fuse is lit prior to the the rotational point of chambering and firing, the last 3 rotational points are inside a secondary (ignition) chamber aligned on the ammunition's arc such that debris from the fuse's ignition can be cleaned out in a separate, external part. This 'external' arc is open to the air forwards, allowing any accidental explosion or pre-ignition due to dust build-up that may occur to be directed away from the user and his allies(if not quite constituting a second barrel)
Depending on the design, the automatic motion normally chambering a round could be aligned to, say, every fourth or fifth (or less or more depending upon tolerances) fire a spring loaded wad instead of a bullet. There is after all no reason why a gun need be cleaned from the muzzle end of a breach-loaded weapon. (And the wad does not need so much strength as to hit the enemy, just to leave the barrel)
Don't really know how light such a contraption would be.
More from wikipedia -"The grouped barrel concept had been explored by inventors since the 18th century, but poor engineering and the lack of a unitary cartridge made previous designs unsuccessful. The initial Gatling gun design used self-contained, reloadable steel cylinders with a chamber holding a ball and black-powder charge, and a percussion cap on one end. As the barrels rotated, these steel cylinders dropped into place, were fired, and were then ejected from the gun. The innovative features of the Gatling gun were its independent firing mechanism for each barrel and the simultaneous action of the locks, barrels, carrier and breech.
The ammunition that Gatling eventually implemented was a paper cartridge style round charged with black powder and primed with a percussion cap. because self-contained brass cartridges were not yet fully developed and available. The shells were gravity-fed into the breech through a hopper or simple box "magazine" with an unsprung gravity follower on top of the gun. Each barrel had its own firing mechanism."
So we have here an existing weapon that uses things that were available in the 14th century, paper, brass, gunpowder, ratchets, springs, primers, multi-phase automation.
Mass production, precision engineering, chemical uniformity and quality assurance processes do not make a machine gun, they just make good machine guns.