Black powder is a relatively simple thing to put together, with the earliest recorded use by China in the 9th century, and plenty of ATLs in which this or that iron-age civilization chances upon it, like Robertp6165's "The Guns of the Tawantinsuya". Smokeless powder is a different animal, and requires fairly advanced notions of chemistry to develop. Hence this challenge: what is the earliest plausible date for someone to come up with a usable form of smokeless powder, and what would the consequences be?
I'm guessing it probably wouldn't be before the early 19th century at best, though in a TL like Thande's "Look to the West", chemistry seems to have progressed noticeably faster than in OTL in the second half of the 18th century. But even if we posit development around the 1840s (guncotton, an unstable early version of smokeless powder, was invented in 1846), it could have quite an impact for such events as the American Civil War and the Franco-Prussian War.
The ingredients for the first true Smokeless Powder, Poudre B, were...
--Gelatinized Nitrocellulose
--Ether
--Alcohol
Alcohol obviously presents no problem. The other two do.
Nitrocellulose consists of Nitric Acid and cellulose (commonly cotton or some other natural plant fiber). Nitric acid was discovered in the 8th century A.D. However, the early nitric acids were very weak and it was not until the mid 17th century that stronger nitric acids, which would be necessary, were produced in quantity.
Ether was first synthesized in 1540 (it is claimed that it was discovered by an alchemist in 1275, but there is no proof of this).
So the earliest date the various materials would be available would be the middle of the 17th century.
However, they still have to discover the process of gelatinizing the nitrocellulose (guncotton), which is far too unstable and dangerous to work with in it's original form. In OTL, it took about 40 years for that to happen from the time that guncotton was first discovered in the 1840s. Assuming the same time period for that discovery, then we are looking at the opening years of the 18th century as the earliest possible time for the invention of smokeless powder, assuming the discovery of the components necessary has not be advanced by some other POD.
It will quickly be seen that smokeless powder's full potential can never by utilized with a smoothbore musket. So the development of rifles, and a solution to the problem of loading a rifle rapidly, is accellerated.
Now, if smokeless powder is introduced in say, 1700, that could create all sorts of interesting butterflies. The War of the Spanish Succession will be fought at first with black powder muskets, but by the end of the war, smokeless powder rifles will be making their appearance on the battlefield in limited numbers.
Sometime in the interwar period, you probably see something like a Minie ball developed for the rifle.
The War of the Austrian Succession will be fought by armies equipped mostly with muzzleloading, single-shot flint-lock rifles firing smokeless powder and "minie balls".
As a result of the losses suffered in the War of the Austrian Succession, most armies will abandon their brightly colored uniforms and begin adopting looser, skirmishing tactics over closely packed columns and lines.
The Seven Years War (or its equivalent in the ATL, as the original war may be butterflied away by earlier events) would likely see the introduction of a breechloading firearm firing smokeless powder wrapped in a paper cartridge. The invention of smokeless powder, per se, need not speed up the invention of the percussion cap, so these will likely still be flint-locks. The last armies to abandon bright uniforms and linear tactics will certainly do so after this war. You probably also see the beginnings of trench warfare as the increased rate of fire drives the armies underground.
Once the percussion cap is invented in the early 1800s, you probably see the transition to metallic cartridge ammo speeded up considerably...introduced by the mid-1820s, perhaps...with repeating magazine rifles, and possibly early machine guns, in wide use by the 1850s.