"How modern and formidable a military might they be able to produce in six months? A year? Three years?"
Since the goal is underspecified, one can imagine two general approaches:
One can imagine that the experts set up shop somewhere and work on their own, producing some pistols, some rifles, some cannon, a little bit of ammunition, etc.
This may work on a small scale, and it may be a very interesting story. It won't work on any kind of large scale. So it all depends; are we speaking of some small principality, engaged in conflict with some other small principality? Becase if we aren't, this approach is not feasible.
Why it isn't feasible on a large is best illustrated with the example of the German Messerschmitt Me 262, the first jet powered fighter in the world and by far the most advanced fighter plane engaged in combat during WW2. What was the overall military impact of the Me 262? Zero. Why? Because there were way too few of them to make a difference. Yes, when they engaged American planes they claimed kill ratios about 4 to 1. But the Germans only ever made about 1,400 Me 262s, while the Americans made over 12,000 B-17 Flying Fortress bombers, over 10,000 P-38 Lightning fighters and tens of thousands of other types of aircraft. A small force with advanced weapons will be defeated by a large force with not-so-advanced weapons. (The German armed forced of WW2 can furnish numerous more examples of advanced weapons produced in too small quantities to make a difference: V-1, the world's first cruise missile; Messerschmitt Me 163 Komet, the world's first rocket plane; the enormously powerful Tiger II tank; etc.)
The second option is to imagine that the experts set themselves the task of transforming the economic base and the military forces of the principality, propelling their pet state from the late Middle Ages to the Modern Age as quickly as possible. This is the only approach which will work on a large scale, but it has the disadvantage that it takes quite a long time, measured in entire generations.
The first thing to ask oneself is why on Earth there were no "formidable" military powers in Europe in the 15th century?
Oh wait! There was exactly one formidable military power: the Ottoman empire. What's the difference between the Ottoman empire of about 1450 and France or England or Poland or Hungary about the same time? (Those are the only candidates for "formidable" military powers; Germany and Italy did not exist, and Spain was in the process of completing the reconquest of the Iberian peninsula from the Moors and was preparing to self-destroy its economy by expelling the only subjects who actually paid contributed to the treasury.)
The difference is that the Ottoman empire was rich, large, populous and well-governed. They had a functional economy, fully monetised, very much more productive than the sluggish medieval states of Europe, soon to be bolstered by the arrival of tens of thousands of industrious law-abiding tax-paying former Spaniards. (The sultan Soliman the Magnificent is said to have expressed astonishment at the foolishness of a king, Ferdinand of Spain, who willingly impoverished his own state in order to further enrich the Ottoman empire.)
Now, this is critical. At the level of technological development of the late Middle Ages, a regular European state could afford to keep not more than 1% of its men away from work and playing soldier. The Turks, with a larger and more efficient economic base, could afford to maintain 2% to 2.5% of their men under arms. Combined with the larger population of the Ottoman empire, this meant that the Turkish armed forces dwarfed the military capabilities of any European kingdom.
Point 1. The first thing the experts must do is put some vim into the economy of the principality they want to transform. Without a viable economic base, luxuries such as armies and navies are simply unaffordable.
To make the economy more efficient, three things are needed: a practicable legal and judicial framework; transition from the barter-based economy of the Middle Ages to a fully monetised economy; and sufficient available manpower to actually work in the new professions, trades and occupations.
Which brings us to the following points:
Point 2A. The team of experts must include somebody who actually knows about late medieval legal and judicial systems, and can devise a plan to evolve them into something resembling the Early Modern period. This is not trivial. For example, France, Spain or Poland never managed to complete the transition, and had to be dragged by force into the modern age, legally and judicially speaking. (This being the major historical merit of the bloody French Revolution and the even bloodier Napoleonic Wars.)
And this takes time. Laws need to be drafted, considered, and promulgated. Judges and lawyers need to be trained. The people must learn and internalize the new laws.
Point 2B. The legal and judicial framework is not enough. To transition the economy from barter to money, there is an absolute need for money and banks. The team of experts must include an economist who actually has a clue about the history of economy (which is rare, but a handful of such do exist), somebody who knows how to produce coins of constant weight and purity efficiently, and somebody who knows how to mine for and purify gold and silver. And the principality must actually have deposits of gold and silver available, or else they won't be able to pull it off on their own and will depend on raising the general economic level of the continent, which we don't want.
Note the the aristocracy will instinctively fight tooth and nail against the monetization of the economy, because this will render them both poor and powerless. The Prince must do whatever is needed to curb the inevitable aristocratic revolts.
Point 2C. A modern-ish (say, England or Germany about 1750) legal and judicial system, and a fully monetized economy require a literate population. The principality is starting with the abysmal literacy level of a late medieval Europea state; to reach a resonable level of 70% to 80% percent literacy there will be a need for a sustained, long term effort, financed by the state. The experts must include experienced teachers, who can teach other teachers to teach the peasants reading, riting and rithmetic; and, of course, the state must ruthlessly suppress the natural tendency of the aristocracy to oppose teaching the peasants.
Point 2D. The real catch is how to make manpower available. What is needed is more efficient agriculture on one hand, and the first rudiments of the industrial revolution on the other. The experts must have somebody who really knows how agriculture progressed between 1450 and 1750, has enough practical experience, and can teach others. Such people are rare, but they do exist. There must also be people who actually know how to make a spinning jenny, a primitive powered loom, how to make paper in the style of the early 1700s, how to make printing presses of about the same time, etc.
Oh great, we have worked flat out for about 20 years and what we have is something sort of like France around 1700, but with a better legal and judicial system. No military power house yet, not an industrial power house yet, but with the premises of becoming one, only if...
Point 3A. The experts must now concentrate on developing the industry, and step 1 is introducing uniform and unchallenged weights and measures. This is hard, and they must rely on the unwaivering support of the Prince, who is likely the heir of the original Prince which started it all at Point 1. Their group must include a metrologist well versed in the history of the discipline, who knows how uniformity of weights and measures was achived without the need of the bloodshed of the French Revolution.
Point 3B. By now the principality should have a decent iron production with purely artisanal methods. It is time to industrialize it, and this means coke, and large blast furnaces, and finery forges. The expert metallurgist with a historical penchant comes into their own; not only making lots of iron and steel, but also lots of bronze for the cannon. It's time to introduce large scale manufacturing of modern-ish firearms, such as flintlocks, and the fabled Minié balls.
Point 3C. And don't forget the transportation infrastructure: large armies need good roads. The civil engineer among the experts finally gets to design and build roads and bridges, and possibly the occasional canal.
Point 3D. And don't forget fortifications. Your progress in artillery will soon be duplicated by the enemy; hopefully, the team of experts includes a diligent student of Vauban who knows how to design and build fortifications in the style of the early 1700s.
Point 3E. During this time, the army grows, training improves, and the principality can introduce late Renaissance tactics such as the combined pike-and-shot formations made famous by long wars between Spain and the rebellious Netherlanders.
Finally, about 40 years from the beginning of the adventure we have the economic base and the manpower to begin thinking about the large-ish armies of 18th century Europe, and the Prince is allowed to think of conquest and everlasting military glory.