Guns in medieval times had been used in battle for at least a hundred years by the time of the Battle of Bosworth. Handgonnes, or hand cannons practically were probably used in the battle as shot around 20mm had been found. These were still early days for hand held guns and they had just barely evolved past being a barrel tied to a stick. Metallurgy too was pretty miserable, as the fragments found at the Battle of Towton suggest. Gunpowder was also still relatively problematic, as the mechanical mixtures would settle and change their proportions till ill effect.
But things were changing around the time of the battle. Corned powder was making it's way around Europe at that time, solving the mechanical mixture part of gunpowder, as the gunpowder was pressed into cakes and grains and dried. This solved the problem of it's constituent parts seperating, but now it was more powerful, straining the metalurgy of the cannons to the breaking point. For a time, all corned powder was saved for the relatively small handgonnes, and the cannons used the old powder.
Handgonnes were changing too. A simple matchlock mechanism was being added to the hand cannons to fire them. This consisted of an s shaped rod, the center pinned to the gun, and a match was tied to the end. To fire the gun you rotated the match into the touchole. These guns later became more complex and slowly evolved into trigger systems that we would recognize today. The guns, while clumsy to load, could still penetrate a knight's armor of the time. This ultimately led to the little guy on the battlefield being just as dangerous as the rich guy in the armor.
A handgonne of the type that could have been at the Battle of Bosworth is available from the Rifle Shoppe. as a barrel in .62 and .75 caliber, and the match arm is also available. You have to supply your own stock and build it yourself, but it is a simple affair. The maker claims it is pretty accurate also.