The Anglo-Saxons had a system where each unit of land (called a hide) was responsible for providing a certain number of men for a local army (called a fyrd). The details of who exactly was called up for this army are not very clear, nor is anything known (as far I know) about their training. At the beginning of the poem The Battle of Maldon there are a few lines where the commander Byrhtnoth, who was the ealdorman (basically the earl) of East Anglia, had to instruct the men how to hold their shields properly and stand in a shield wall. This could be basic poetic description or it could be referring to actual military instruction on the field of battle. There are tons of books and articles on the fyrd which googling will turn up for you, but a good one is Richard P. Abels, Lordship and Military Obligation in Anglo-Saxon England.

As far as I know, however, there was no such thing as a school for training peasants to be like knights and such a thing would be so antithetical to a medieval knight. Peasants were not knights and the separation between them was far deeper than simply not having been taught how to fight. Their entire reason for existence was based on a firm division in the members of society. This division could be expressed in many ways, but a common one is the threefold distinction of those who fight (knights), those who pray (clergy) and those who work (peasants). There could be some cross-over between peasants who were raised in the church, but I don't think there was a path from peasant to knighthood. There is a Middle English poem called Havelok the Dane where a nobleman is raised among peasants but is better than them in every way. He looks better, is stronger, faster, and so on (plus a light shines from his mouth when he sleeps in case anyone wasn't sure just how much better he was than everyone else).

So my answer would be no, something like this would never be in the realm of possibilities, but don't let that stop you from writing. Make up a king who takes a chance on it. Maybe he will be murdered by the other nobility who think that his ideas are destroying the honor of knighthood. Or make up your own country where such a thing is possible, or even set it in your own world. If the story and characters are fictional, why does the world have to be real?

My limited historical research was on identities of the peasantry in 16th Century Sweden. From that research I would like to point out that your notion of laying down ones life for the country is a bit anachronistic.

The Peasantry had numeroes identities as anyone today would (country, region, religion and so on) but the notion of a strong national identity is rather more modern. See the great work of Benedict Andersson in his very influental book, Imagiened Communities.

Regarding institutions of secular training in medival Europe see The first European Revolution by R I Moore. It deals with issues of training a civil State and the strugle between Church and secular authority.

I will answer for the later Middle Ages, because that is my specialty.

By the later Middle Ages there were many people who fought as knights but did not have the title. Those who fight as knights, with and without the title, are referred to as 'men at arms' in English or 'gens d'armes' in French (from which comes the modern 'gendarme'). Judging by the discrepancy between the number of people with the title 'knight' in late Medieval England and the number of men at arms fighting in various battles, it is likely that -most- men at arms were untitled in 15th century England.

However, these people were not by any means poor. They were expected to provide their own arms, horse and equipment at a bare minimum (the wealthier among them would be contracted to provide retained archers as well). Since a full harness (suit) of plate armour cost around 100-160 days wages for a common soldier for much of the late medieval period, the armour alone would be prohibitively expensive for someone that was not significantly wealthier than average. Add in the cost of a warhorse, a spare horse, and everything else, and even your 'less wealthy' man at arms is significantly better off than the median Englishman. And indeed, many men at arms styled themselves 'Esquires' or 'Gentlemen' (a more nebulous concept than 'noble'). Medieval society was not simply divided between titled nobles, the clergy and a mass of 'peasants' - many people lived in a space in between (the Paston family in England is one famous example - they start off as ploughmen and end up as knights within 4 generations, via becoming very successful lawyers).

This all hints at the big reason why your proposed school of knighthood didn't happen historically. Medieval Kings in Western Europe did not have the tax base to handpick, equip and train their own personal soldiers. They relied upon assembling armies as needed on a per-campaign basis. The system for doing this varied across time (in the later Middle Ages in England and Burgundy this was done through a system of contracts). But a medieval monarch didn't have the kind of sustained, reliable resources to make this kind of project economical. It was much more cost effective to hire or call upon an existing class of semi-professional soldiers (this applies to say, Swiss Pikeman and English Archers as well as men-at-arms).