Specifically, around the 11th and 13th centuries, what could we assume a peasant, living in medieval England would have known? Perhaps one living in a farming community inhabited by other peasants, a lord, and perhaps a church head as seen at Wharram Percy. Would they have any knowledge about the Roman Empire that existed a millennium before? Would they have known who the ruling king or Pope would have been? Or did the extent of a peasant's knowledge only involve agricultural techniques and animal husbandry?
Thanks!
Your major underlying assumption is that we can classify a whole class and "type" of people as generally knowing or not knowing things. Medieval peasants were as diverse in their intelligence as people today are; you'll still find college graduates in the U.S. who can't name a single congressman or locate England on a map; you'll meet "uneducated" people that are more versed in Plato than some Ph.D.s.
Here's what we can say: Medieval peasants were not averse to travel; many would have geographic knowledge and familiarity with different regions/cities. They had to pay taxes; they probably would have known who they were paying taxes to. They went to church every once in a while; they would have had some knowledge of theology, saints' lives, and church leaders. Everyone's life is historical: a medieval peasant would have known whatever relevant historical information they felt they needed to fulfill personal interests or form an identity, whether that be Ancient Rome or whatever.
A peasant who felt a personal draw toward his/her christian faith would know more theology and popes than one who didn't really care that much and was content to let the priest do the work. The peasant who wanted to get a leg up by increasing profits for his agricultural production would obviously learn more about markets, business, and places to buy/sell. Obviously, a medieval peasant didn't know about germ theory; but if knowledge was available during the time, there's a chance they would have known it. Certainly knowledge about the Roman Empire was available to someone who wanted to seek it out.
I can't give any examples from 12th century England but there are some from a little bit later and further south. There's the Disputation of Mallorca, which records an ongoing dialogue about religion between a Jewish merchant from Mallorca and a middling Genoese merchant. The Genoese merchant is able to elaborate on Christian doctrine and belief in ways that are relevant to the discussion though they're not as complex or well-argued as some of the highly-educated church apologists were. He had extensive knowledge of the bible and Church doctrine, if it was a little simplistic. There's also the famous case of Menocchio from 16th century Italy, published by Carlo Ginzburg, where a peasant mill worker is recorded elaborating on a complex worldview that blended church doctrine, folk beliefs, and his own thoughts and feelings.
The point is, peasants were individuals and functioned as individuals rather than a class; they're hard to stereotype in terms of something so personal as "knowledge".
Would a peasant in 170 AD Rome be better educated than a peasant in France in 1200?