I’ve been served a fair number of almond-based medieval sweets and trifles, (and some meat & almond milk dishes that I dislike) but I honestly never thought about the historical provenance of almond milk before - good question!
I do know that both almonds and rice traveled the Silk Road and were very well known throughout Europe and the Near East in antiquity, well before the medieval era. Almonds have been cultivated for something like 5,000 years at this point, and were known in England before the White Christ.
Anyway Gerard’s Herball references the milk, fruit and leaves of the almond tree frequently, in a way that implies that by the 1800s literally everyone who could read was intimately familiar with almonds and their products and processing; he says stuff like “the fruit of such-and-such is divided like the almond” and “a juice or liquor is trodden from so-and-so in the same manner as milk from the almond” &etc.
Wikipedia references Holy Feast and Holy Fast: The Religious Significance of Food to Medieval Women by Caroline Walker Bynum to claim “Almond milk was a staple of medieval kitchens because cow’s milk could not keep for long without spoiling.” I can believe the first half of that sentence (as Wikipedia also notes, in the Medieval era “almond milk was known in both the Islamic world and Christendom… [and] …it is suitable for consumption during Lent”) but I have a little more trouble with the second half of that claim, since producing and preserving dairy foods was a solved problem in the Middle Ages.