[Part 1]

Sorry for really late response.

Tl;dr: While I neither expect all of the medieval rulers knew as much as legendary King Arthur did in cited scenes, at least a few of them who especially lived after the 12th century might be able to know better and compose a even more knowledgeable answer than him.

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(a) how informed were the experts of the day in how birds worked

Seasonal migration of some species of the birds, including swallows, had in fact been observed for long, since Roman period.

Gerald of Wales (Giraldus Cambrensis) (d. 1223) comments on the migration of stork as following:

"Chapter XV: Of birds which disappear during the winter. It is also remarkable in birds of these [storks] and other similar species, which the rigour of winter is wont to drive away, that during this period they are neither living nor dead, but vegetating, without the breath of life being extinct, they appear wrapt in a long trance, and, remaining without the nourishment by which animal life is wont to be sustained, are yet supported by some kind and secret process of nature, until, roused from their sleep, they come back with the zephyrs and the first swallow (Topographia Hibernica, I-15. The translation is taken from [Forester & Wright trans. 2000: 23])."

While Gerald is totally wrong to assume the alleged hibernation status of storks (my another favorite mistake in his writing is his claim that barnacle gooses (Branta leucopsis) don't breed naturally, but in fact grow from fir timber in as many as a thousand birds [like fruits] - he also claims to witness its growing in person!) (Topographia Hibernica, I-11), at least he know some species of the birds are migratory, and he also quotes his concluding sentence of these cited passage in fact from Roman Poet Horace's letter (I-7), ".....you, dear friend, he will revisit with the zephyrs, if you will give him leave, and with the first swallow (hirundine prima) (The translation is taken from Guthenberg.org)."

As all of we know, Horace was not the scientist (I suppose it was a kind of seasonal greeting), and the point here is that even his "the first swallow" phase presuppose the basic knowledge of seasonal migration, or at least seasonal appearance/ disappearance of swallows. According to late Karsten Friis-Jensen, Horace's poems (including letters) were widely used to teach Latin in the cathedral school in post-Carolingian medieval west (since the late 10th century) (Friis-Jensen 2007: 293). Thus, it would wonder little if some decently educated middle to high rank Christian clergy (like those who also often serve the ruler as a chancellor), not necessarily a limited number of dedicated scholars (or "scientists" of the period), would have heard of the quote in question.

(b) did any of this information make its way to the royal rulers, and was it considered useful at all

Conveniently enough, Gerald was also a royal chaplain to King Henry II of England (d. 1189) and dedicated his Topography of Ireland (Topographia Hibernica) to him. So, we can reasonably suppose that Henry II also knew some birds like swallows were certainly migratory.

On the other hand, however, we are also sure that there was in fact at least one medieval ruler (though I suppose at least one more) who must have known much better even than Henry II - he was indeed also the first and foremost pioneer of the ornithologist (a kind of) in medieval Europe!

So, now it is time to cite some passages from Emperor Frederick II [of HRE (d. 1250)] 's the Art of Falconry (de arte venandi cum avibus) to check his knowledge on the migration of birds - in fact, this alleged treatise of the art of falconry in tremendous 6 books includes the very detailed description of habitats of diverse bird species in its the first book, based on his own empirical observation as well as written works like Aristotle's History of Animals (Wood & Fyfe trans. 1943).

Frederick even criticizes some of Aristotle's writings based on his own observation, in the prologue of this work:

"We have investigated and studied with the greatest solicitude and in minute detail all that relates to this art [falconry], exercising both mind and body so that we might eventually be qualified to describe and interpret the fruits of knowledge acquired from our own experiences or gleaned from others..... Inter alia, we discovered by hard-won experience that the deductions of Aristotle, whom we followed when they appealed to our reason, were not entirely to be relied upon, more particularly in his descriptions of the characters of certain birds.

There is another reason why we do not follow implicitly the Prince of Philosophers [Aristotle]: he was ignorant of the practice of falconry an art which to us has ever been a pleasing occupation, and with the details of which we are well acquainted. In his work, the Liber Animalium, we find many quotations from other authors whose statements he did not verify and who, in their turn, were not speaking from experience. Entire conviction of the truth never follows mere hearsay (Wood & Fyfe trans. 1943: 3f.)"

To give an example, Frederick observes the swallow catch flies in the air, in contrast to the majority of other bird, and writes as such:

"Certain land birds take their food on the wing, others on the ground. Some (for instance, swallows and siskins) devour their prey in the air. They catch such insects as flies, beetles, bees, wasps, and other flying insects but avoid devouring in the air those provided with stings, which might wound their digestive apparatus. Such insects are generally killed and prepared for deglutition either on the ground or while the bird is roosting on the branch of a tree so that they may be ingested without danger (I-9, in: Wood & Fyfe trans. 1943: 20)."

As for the migration of the birds, Frederick indeed spends 9 of the total 57+ chapters in the book 1 of his treatise. The most remarkable point is that he classifies diverse degree of migration between latitudes in accordance with the concept of 7 climatic zones theory in medieval Islamic science:

"The localities to which birds migrate (mainly to escape the cold weather) and in which they plan to spend the winter are of great variety; some are remote and others less distant.

Birds that nest in far northern latitudes or in regions within the seventh climatic zone are, as a rule, satisfied to migrate to the sixth or fifth zone, but may in turn abandon this region to go farther south. Birds born in the seventh climatic zone do not remain there throughout the year; if they reside continually in their birthplace they must not be classed as migratory birds. Birds migrating from the seventh to the sixth or fifth climatic zones (or beyond) find in these latitudes warmth sufficient for their needs in contrast with the intense cold of their native land from which they are retreating. The same holds true of species born in the sixth and fifth zones, the majority of whom are content to travel no farther south than the fourth and third regions, where the climate is mild enough for a winter residence. It is the same with birds born in the fourth and fifth zones; they migrate, as a rule, to the second climatic zone, or still farther south (I-23, in: Wood & Fyfe trans. 1943: 40f.).

In this theory, seven climate zones divides the globe by latitude.

(c) how else were leaders expected to understand science?

One can argue against my long citations and argument above that Emperor Frederik was just an very exceptional, isolated case during the Middle Ages - Frederick was indeed called as "the wonder of the world (stupor mundi)" by contemporary authors and his knowledge was based mainly on his motherly cultural heritage in the kingdom of Sicily. His writing also had attracted very little attention from early scholars until the 18th century.

It is worth noting, however, that King Manfred of Sicily (d. 1266), son of Frederick, also wrote a few additional section on habits of birds to his father's treatise in one manuscript. So, at least the son of the Emperor shared the interest as well as a part of this kind of knowledge with his father.

I'd also suggest that Frederick himself also regarded this kind of knowledge primarily as this kind of knowledge of birds not just as "pure" science, but also as a part of medieval falconry, arts for noble rulercraft. As I mentioned before in: Were gifts between kings supposed to be useful?, falconry was a very popular art of pastime among the elites across Eurasia, and exchanges of gyrfalcons and other exotic animals like polar bears sometimes bore a diplomatic significance.

[Continues to be the part 2].

(Edited): fixes typo.