Matt Easton over at the Schola Gladiatoria gets mentioned a lot in regards to discussions of Western martial arts, and for good reason. He is very cautious with his words and is a trained archaeologist, hobbyist sword collector (and dealer), WMA school owner and competitor. His Youtube channel is worth a perusal if you want to explore this topic.

I would recommend the following video, in a rare turn of events I would also say that he gets some very informative youtube comments as well:

http://youtu.be/MMl5tyO5fC8 - A discussion of how fencing treatises are not for the battlefield but how they might relate.

Matt Easton often mentions the book Swordsmen of the British Empire by D.A. Kinsley - I own a copy and it is mostly a collection of letters and first hand accounts of sword combat during the 17th to 20th century with most of the focus being on the 18th and 19th - so it is a little outside the scope of late medieval, but it does allow me to touch upon an important point that is often covered on Schola Gladiatoria: swords were primarily a sidearm for the history of their use. Their real utility was in being able to be carried conveniently, a weapon to have on you when you're caught out.

The scenes set in Swordsmen of the British Empire often (but not always) take place outside of large scale battles, where one person is faced by one, to a few, enemy combatants. In these situations, which this book presents as not uncommon in war, even with the professional army seen in later centuries, a fight between two men using their sidearms (swords) can often play out like a duel.

Essentially, the training does not translate directly but is still applicable in ways.

Edit: I thought about it a little more and feel like you should read George Silver's "Paradoxes of Defence" (1599, still out of scope - sorry I'm not that familiar with the medieval period) which is pretty influential in some WMA circles.

There is little technique in the book (although I've heard some WMA circles teach Silver techniques, not sure how they reconstruct it) and most of it reads as musings on the application of dueling (specifically Italian masters in London whom he disagreed with vehemently) in an actual fight. George Silver is often referred to as a teacher for the battlefield, whereas much of what he critiques is clearly not.

It is also worth mentioning that he ranks weapons hierarchically and rapier, rapier and dagger, sword, sword and dagger, sword and buckler and sword and targe are at the bottom of the list. He clearly favors polearms when it is possible to fight with polearms, further evidencing the sword as a weapon of last resort.

My study in this field was pretty brief (I ended up focusing primarily on ranged warfare rather than melee). That said little I found in the materials on medieval swordsmanship (training manuals and the like used by the Association of Renaissance Martial Arts for example) pretty much exclusively applied in the situation of the late medieval duel. Lots of these tactics related either to one on one combat or else to more tournament style small group warfare rather than to the medieval battlefield. Based on your OP this might not be news to you, though.

One thing to bear in mind when talking about medieval warfare is that the nobility actually represented a rather small proportion of the actual army. If we take Falkirk for example (which in fairness is a bit of an anomaly) Edward I's host was roughly 30,000 men of which maybe 2-3,000 would be armed nobles/knights/squires. Now once the Black Death hit the halving of the population of Europe overall army hosts got quite a bit smaller so the 'elite' would probably become a larger proportion of the army as a result. However, on the Agincourt campaign half the army were archers, probably free peasants, and the other half certainly wasn't entirely composed of the nobility. The peasants or even trained mercenaries used by many medieval armies probably weren't reading the dueling manuals of the elite.

Also worth considering: medieval battles lasted a while. They could go on for hours and even if you started out using elaborate martial arts techniques by about hour 3 of fighting in full plate in the mud you probably are just going for "hit guy with sword, don't get hit in response." That's not to say that knights weren't well practiced or to suggest that they were simply thugs bludgeoning each other with weapons. It's more to suggest that we shouldn't imagine these knights as Jet Li style martial artists.

Edit to add references! Jean Froissart is a good accessible primary source for the battles of Crecy and Poitiers. Crecy is especially noteworthy because Froissart basically directly copied the account of Jean le Bel (plagiarism not a real concern in the Middle Ages) who was actually at the battle and fought for the English there.

Michael Prestwich is great for information on Falkirk. Especially relevant here is his book Armies and Warfare in the Middle Ages.

For infantry warfare in general I quite like Kelly DeVries Infantry Warfare in the Early 14th Century.

For Agincourt and archery: Jim Bradbury The Medieval Archer.

Most of the early manuscripts were found in "commomplace books" or hausbuch which are more like scrapbooks detailing the yearly liturgical events, house-wife cures, bits on astronomy and legal/tax rules, etc. The oldest known fencing treatise was the I.33 - for sword and buckler. There is no indication who or what the target audience is but the illustrations depict priests and students. There is not enough evidence to suggest it was a military order (or any evidence whatsoever, to be honest).

Training methods of medieval armies, which is what you're asking about, is largely undocumented as yet (that we know of!) but as standing armies of a single logistical train did not exist, all matters of training and administration would have fallen on the company captain (or local militia captain, etc). How they would have trained is, again, undocumented but basic pedagogy seems to indicate that wrestling was popular everywhere where there were people and fencing guilds popped up in every city large enough to support a militia. The oldest of the lot is the Brotherhood of St Mark (Marxbruder) and they were around in the mid 1400s.

Fencing guilds certainly existed in even during the early stages of the Hundred Year War (specifically 1369-1370) according to a historical account by Ann Wroe's "A Fool and his Money" - it mentions in passing the presence of fencing guilds as attended by the citizens of the Bourg (a 'shady' part of town) and a separate one for the local lord's sergeants that also kept the peace as a sort of a proto-police force.

You should try asking on /r/wma as well.