I found this article that really is the only one I could find that goes in-depth about the gun itself.
In the above diagram, we see the workings of the Cookson repeater. The bullets and powder can be loaded via a flap on the left side into the two compartments A and B. Compartment A holds the bullets and B holds the powder. The round piece C is the revolving breech block and it has two cavities D and E within it, to hold the powder and the bullet ball. When the breech block C is revolved, a ball drops into the first cavity D and the powder falls into the second cavity E. Note that the front of the cavity E has a diaphragm G in front, which divides the cavity in half. This diaphragm is there to prevents a bullet from falling into cavity E, as the width of the entrance to the cavity is too small for the bullet to fit. As the breech block is revolved all the way around, the ball and powder drop into the rear end of the bore at F. Some of the powder also dribbles into the firing pan. Moving the lever further also cocks the flintlock and closes the pan's cover (the frizzen) and the weapon is now ready to fire. The whole process of reloading a new shot only takes two or three seconds, giving its user a huge advantage over other firearms of that era.
This is a really good description of how the firing system works. I know that there were other repeating-flintlock mechanisms in the 18th century that were supposed to be revolutionary but didn't take off due to mechanical problems (i.e. Puckle gun, Ferguson rifle), but the article doesn't mention any disadvantages or problems with the gun, so I'm wondering why it was still never widely used/distributed.
Here's some other information I find interesting:
This gun appears to have made its way to Maryland, probably brought by one of the early British colonists to America.
Incidentally, there is a record of another John Cookson, a gunmaker in America. This John Cookson is also known for making repeaters and is known to have lived in the city of Boston between 1701 and 1762. Some authorities say that this John Cookson was related to John Cookson from London and others say that he was the same man and had merely moved from London to Boston. Regardless, in 1756, he published an ad in the local newspaper, Boston Gazette, advertising his nine-shot repeaters. He is also known to have made some seven-shot repeaters as well.
The above repeater was made around 1750 by John Shaw, who lived in London and Boston. He is reported to have died in Boston under rather strange circumstances.
Do we know more about John Cookson and John Shaw? I searched them up but they don't have any wiki articles or other biographies.
It is often assumed that breechloading small arms did not exist before 1800, due to a lack of imagination by early gunsmiths or their inability to fit things together. They did exist, and some gunsmiths made them. The Cookson ( or Lorenzoni) design was not more widely spread because it was too modern: it was not more widely spread because it was , like many breechloaders of the time, quite expensive, somewhat unreliable and possibly risky. It required an enormous amount of careful hand-fitting to get the cylinder perfectly fitted to the receiver, and that tight fitting was also a weakness, as gas pressure from the gunpowder of around ten thousand pounds per square inch has a way of getting into joints, crevices. When it did, it would tend to leave a lot of dirt, as black powder converts to quite a lot of solids. That would gum up any tight-fitting mechanism. If the mechanism was loose, to allow for fouling, the gas might get further, getting into the magazine area and blowing up the gunstock and the owner. The general expense, unreliability, and danger of the guns made them unfit for military use. The British Ferguson rifle , along with the American Hall, sidestepped some of the problems by using a mechanism that was not badly fouled and didn't have to be tightly sealed- excess gas could vent away from the shooter. But they were still expensive.
By the US Civil War, breechloading systems had been developed that sealed the breech without also making the gun likely to seize up from powder fouling: the Prussian had their famous Dreyse needle-gun, the Sharps had an internal gasket. The Industrial Revolution had also brought improved machining methods than a gunsmith with a file and hacksaw, and so the more complex breechloading mechanisms also became lower in cost. When the modern brass cartridges definitively solved the problem of sealing the breech, muzzleloaders went obsolete.