Here are some semi-random quotes. I do not have time to chase the references, but they are coming from a modern professional historian, not from a You-tube personality, so I'd take his numbers seriously.
Robert A. Houston, "The Growth of Literacy in Western Europe from 1500 to 1800".
Houston is a professor of History at St.Andrews and wrote a book on the subject:
"Literacy in Early Modern Europe: Culture and Education, 1500–1800," London 2001.
Literacy is a relative concept that has meaning only in specific economic and social contexts, but historians tend to rely on universal, standard and direct indicators such as the ability to sign one’s name on a document. Using this measure it is clear that there were social distinctions in the ability to use a writing instrument throughout the early modern period.
At the end of the Middle Ages, the ability to write was restricted to less than 10% of men and hardly any women possessed it.
The change in literacy rates was halting and irregular. Judged by signing, the most pronounced early expansion occurred among the middle and upper classes, among men and in towns. In northern England the illiteracy of the gentry fell from about 30% in 1530 to almost nil in 1600, but that of day labourers stayed well above 90%.
You can find more numbers in his article (in particular, numbers in relation to different areas of Western and Central Europe). Yes, the ability to read is different from the ability to write, but in Houston's statistics he focuses on the ability to just write own name, which is the bare minimum of literacy and (IMHO) probably was highly correlated with the ability to read (at least in some language, be this Latin or vernacular).
Edit 1. Regarding Mark Olson's request, below are estimates of literacy at the end of the medieval times/early modern times from other authors.
All the books that I found which deal with numerical estimates of literacy in late medieval and early modern times favor the ability to sign as an estimator of the degree of literacy. In addition to Houston's book (whose book has the entire chapter comparing different methods, including ones which I find rather exotic, such as materials of Spanish Inquisition trials):
D. Cressy, Literacy and the Social Order. Cambridge University Press, 1980.
R. O'Day, Education and Society, 1500-1800: The Social Foundations of Education in Early Modern Britain. Longman, 1982.
Adam Fox, "Oral and Literate Culture in England, 1500–1700." Oxford University Press, 2000.
Cressy is especially thorough with providing numerical data and its statistical analysis.
Day compares for instance the analysis of literacy in Northern England in
Jo Ann Hoeppner Moran. "Literacy and Education in Northern England, 1350-1530: A Methodological Inquiry." Northern History, 17 (1981): 1-23.
based on estimates of the number of parochial schools and argues that Cressy's methodology is provides more reliable estimates.
On page 20 of her book O'Day says that Moran's numbers indicate that 15% of the total population of York diocese had undergone basic schooling by 1530, while Cressy indicates 10%.
O'Day characterizes as "wildly optimistic" the estimates of 30% literacy in the late 15th century.
Moran's article is the only one that I found which provides numerical estimates of literacy in England in 13-15th centuries. She notes, however, lack of extant records and bases her estimates on availability of schooling (making the numbers less reliable, at least according to O'Day).
Some of Moran's research results are summarized by O'Day as follows:
From this source [availability of schooling] she concludes that some 15% of the population of York diocese attended a school in the late 15th century, as compared with perhaps 9% in the early 15th century and some 4.7% in the late 14th century. But it may be suspected that these figures err considerably on the generous side. Dr. Moran based these estimates on calculations which assumed a regular and constant size for the schools concerned. Later evidence suggests that consistency in this respect was not a feature of early schools.
From reading Cressy's book, I think, I found the original source of the claim that about half of English population by the end of 15th century was literate. He attributes the number to Thomas More:

Cressy then proceeds:

While Cressy's book contains mass of numerical data related to literacy (based on various archival work), it is all broken into subcategories according to occupation and geography and, in the book, I could not find aggregate overall numbers. But, in his later paper, from 1993, "Literacy in context: meaning and measurement in early modern England", Cressy writes:

Thus, according to this estimate, the rate of literacy in mid-16th century England was about 12.5%.
Another estimate (dealing with the end of the medieval time and, thus, closer to the OP), based on the signage records appears in
Adam Fox, "Oral and Literate Culture in England, 1500–1700." Oxford University Press, 2000.

Based on the early arguments from Schofields 1973 article (which is widely cited and described as the breakthrough leading to transition from qualitative to quantitative estimates in literacy during early modern times), Fox then estimates that these numbers should be doubled to get an estimate of the reading capacity, thus, getting:
10% adult male reading and 2% adult female reading in England in 1500.
Lastly, regarding criticisms of usage of signage as a mean of estimating rates of literacy (ability to read) in late medieval/early modern times:
In their books O'Day, Cressy and Houston analyze in detail other available methods, note merits and demerits of each and, in the end, conclude in favor of signage. (This again follows Schofield's 1973 article.) In particular, they do discuss arguments similar to the one in Adam Baker's comment below, suggesting that for some people signing with a symbol/picture instead of the written name was a choice rather than the result of illiteracy. Reproducing these arguments here would take too much space and my answer is already too long, so I will refrain from doing so.
I am unsure about the origin of the claim that (generically) women were prohibited from "signing anything." I do not exclude that this was indeed the case in some places and at certain times. However, as fas as late medieval/early modern England goes, the claim is plain wrong. For instance Cressy documents striking difference in signage among women in London and elsewhere by the late 17th century: The percentage for both categories started near zero in 1500 and increased only to about 20% outside of London by the late 17th century. At the same time, in London, it reached about 50% by late 17th century. As far as I can tell, this dynamics completely invalidates the conjecture that women (at least in the discussed time period) were prohibited from signing their names.
Ditto the numbers provided by Houston in his article and the book for other parts of Europe: They indicate slow (but uneven) growth of female literacy (as measured by name signage) elsewhere in Europe, indicative of improvement of education rather than abolishing of some laws prohibiting signage.