Part I: Why this is a tricky question

The proportion of soldiers who were in armour in combat is wrapped up with a lot of other questions, such as the composition of armies in the Middle Ages and the equipment of different sorts of medieval soldiers. In particular, the question of the armour of the 'the common soldier' (those soldiers who were not part of the fully armoured social-military elite) is one of the most persistent and vexing questions of the study of arms and armour and its intersection with military history.

Compared with common soldiers, the equipment of the military elite - knights and those who fought as knights - is simpler to describe. I will call these soldiers by the English name, ‘men at arms.’ In late medieval Western Europe these were equipped with some kind of ‘full armour,’ - what this entailed depends upon the era, as armour transitions from mail to plate armour in the period from 1300 to 1400 and beyond. Depending on the situation, men at arms might not wear full armour - when fighting on foot in certain circumstances, or in a siege, for instance. But they would be wearing metal armour. So we can say that in a late medieval army, all of their men at arms would be wearing mail or plate armour.

This leaves those pesky common soldiers.

This question is made more difficult because there is no 'average late medieval battlefield.' Each battle is different, each war is different, each country is different and each nation is different (not all of Europe is England - keep that in mind when I quote English sources below). Different soldiers equip themselves differently, reflecting their way of war, their economic means and the larger technological world they live in. And that world changes a great deal over time. The period that you mentioned is a very long time in the history of arms and armour, which covers plate armour being a supplement to mail in 1300 to plate armour being the primary means of defense in 1500 (and defence for an increasingly large number of people, as well!).

A difficulty is that we don't have a complete accounting of the equipment of a medieval army. We have a lot of other sources - we have the equipment of -parts- of armies in the form of the inventories of royal arsenals, the estates of nobles and the muster rolls of militia, but this isn't a total picture. We have many drawings of medieval soldiers in battle, and depictions of them in other contexts (some contemporary, some historical or fantastical). Often these show many people in full armour but we consider such pictorial sources carefully - medieval artists did not necessarily draw photo-journalistic records of 'the way things were' but used a variety of visual tropes to convey their intended meaning. This meaning could be 'this man is a soldier and soldiers are armoured,' rather than 'this is what a soldier of my time and place looked like, on average.' The final source that we have in this period is the ordinances of princes (namely Charles the Bold of Burgundy) who were setting out the equipment for their soldiers. While we know these were guidelines, we don't know how often they were followed - if they were exceeded, or if soldiers were hired even when they didn't meet the minimum.

Because of the incomplete and fragmentary nature of our sources, a definitive quantitative answer is not possible, particularly one that covers such a large period of time and space. We cannot say ‘on average this percentage of people would wear this type of armour.’ But that doesn't mean that we can't say anything! Instead of stating exact percentages we can make some observations about the evidence we have and point out some general trends.

First of all, there is the question of how armies were equipped. It is important to keep in mind that medieval armies comprised soldiers who were raised in different ways. Generally, in the later middle-ages armies were largely made up of semi-professional soldiers recruited, retained and paid under some contractual system - such as a mercenary company, an individual contract with a sovereign at war, or a contract as part of a lord's retinue. In some circumstances, soldiers were drawn from local militias*, but one shouldn’t overstate the importance of these - since they were called up in the face of imminent threats, they weren’t suitable for anything other than defensive battles, and mostly local ones at that. While the old saw that 'medieval soldiers bought their own equipment' has some truth to it, that is not the whole story. I go into this more this previous answer:

By and large, medieval soldiers were supposed to supply their own armour and weapons. Those who served were, generally, those who could afford to arm themselves - landholders, both the aristocracy and the wealthier commons (and keep in mind what ‘common’ means is somewhat fluid across time and place), and substantial townsfolk (who become important as soldiers in certain regions, like Flanders and Italy). Medieval Kings, even on those occasions when they raised troops by levy in the high middle ages, were not rounding up peasants from the field and putting spears in their hands - they were calling upon, or hiring, the classes of society that were armed (or were -supposed- to be armed). This largely remains the case, when rulers in Northwestern Europe use contractual soldiers in the later middle ages, which was their primary way of raising armies in that period. In cases where soldiers provide their own weapons and armour, if war came there would theoretically be no need to create new weapons and armour, because the soldiers would show up with what they already had. However, there might be a rush to purchase weapons before a major war or campaign as soldiers updated equipment or struggled to meet their obligations to provide it. However this would be disorganized, since it would be individual soldiers (knights and infantry both) doing the purchasing. It is best to think of individual soldiers providing their personal weapons as the -primary- means of equipping soldiers into the 16th century. However, this could be supplemented by other sources. Another way Soldiers could be supplied was by their immediate employer. English soldiers of the later middle ages, particularly in the later 15th century, were often the retainers of some greater lord - they signed a contract of ‘livery and maintenance’ to serve that master in return for pay, clothing and food. It is unclear how many soldiers supplied all their own arms in this system, but we know that great lords could have substantial arsenals to equip their retinues.

Finally, armour and weapons could be provided by a monarch directly. In order to keep and maintain a stock of weapons, kings established royal armouries or otherwise arranged for the direct purchase of armour. I am best acquainted with purchases from the end of the 13th century and later. Through a series of bureaucratic developments, King Edward III’s primary armoury came to reside at the Tower of London. A wealth of records survive from the 14th and early 15th centuries. These inventories sometimes seem trivial in their details - the same 6 great helms show up for several decades -, but general patterns emerge. By and large, the Tower Armoury accepts a large new stock during wartime, and then seems to use it up. So a massive stock of armour (thousands of bascinets, hundreds of pairs of plates, etc) is taken in when war with France heats up in 1338, and then stocks decline as these are distributed. The items that stick around are those that are not actively being used in warfare; those 6 great helms were of little use to late 14th century armies. Eventually they are discarded (recycled, repurposed, thrown in the Thames, perhaps). The armour and weapons purchases of kings could be enormous. As early as 1295 the merchant Frerderic the Lombard collected in Bruges the following items for the Fleet of King Philip of France:

2853 helmets 6309 round shields 4511 mail shirts 751 pairs of gauntlets 1374 gorgets 5067 coats of plates

There is not necessarily a direct correspondence between how soldiers were recruited and how they were equipped. Certainly the arsenals of great lords were meant to equip their retainers, but some (many) retainers may have come with their own equipment. Similarly even local militias in certain places (Swiss and German Cities in the 15th century) might supplement their own equipment with equipment from a town arsenal. Even members of the military elite, the fully armoured men-at-arms, might receive armour as gifts from patrons further up on the social hierarchy; we know that English Knights ordered armours by the half dozen to equip their squires. Edward IV ordered 100 full armours at a time to equip his own men at arms. In general, we can surmise that rural militias, lacking centralized arsenals and patrons, probably relied more on their own armaments, while the retinues of great lords could probably rely more on gifts or issue - but not entirely. Soldiers of all kinds might bring their own armour; soldiers of their all kinds might find their equipment supplemented from other sources. This meant that the cost of equipping oneself was an important limitation on the quality and quantity of armour available but was not the sole factor. The ability of cities, the nobility and kings to organize and distribute wealth and patronage could also equip soldiers.

*I prefer the term 'militia' to 'levy' since it conveys that these forces were defensive and local. As such, they were not simply peasants conscripted by a lord and forced to fight, but local people with a strong motivation to defend themselves, their family and their property.