This question strikes at the core of a problem that troubles anyone who seeks to extract folk belief and tradition from primary sources. Too often - as is the case with Geoffrey of Monmouth - there are no other sources that can lend perspective and dimension to that same time and place, that would give us an opportunity to triangulate on what was happening in the author's world that may have inspired his writing, or to give us insight into how his material was popularly received.
The first step in the process is to evaluate the nature of the author and his text. Tradition holds that Geoffrey of Monmouth was a Welshman born around 1100, but that nationality is disputed. O.J. Padel suggests, for example, suggests that Geoffrey was actually a Cornish man. That's not much of a reach since the Cornish and the Welsh at the time spoke a similar language and were often blurred in the minds of the English as those people "to the west." The point here is not of critical importance, but it underscores how little we know about this medieval author - and it follows - about his intent and fidelity to folk tradition.
The other thing that we must concede when dealing with a twelfth-century British text is that even if the author was true to traditions he had heard (or in this case claimed to have received in written form), there is no way to know how widespread those traditions were or how widely they were believed. There are ample examples of folklorists and ethnographers working with cultures that included "true believers" in various things, and invariably one finds skeptics who don't believe and members of that society who hadn't heard a given tradition. It follows that in part, we can answer your question of "would readers of the time have understood the prophecies to be legitimate and accurate" with a two-fold answer: even if this text was taken from folk tradition, some may have not heard of the tradition and among those familiar with the tradition, some were certainly skeptical. The first point may have been particularly true of Britain in general if the Merlin prophecies were taken from Welsh and/or Cornish tradition, places far removed from Norfolk, Yorkshire, or Scotland. Certainly many would not known of this tradition. Where and if the tradition was active (presumably in the far west), some may have not heard of this tradition, and others may not have believed in the tradition. The manuscript suggests that Alexander of Salisbury, Bishop of Lincoln, asked Geoffrey to translate "The Prophecies of Merlin" into Latin, suggesting that there was a proto-Geoffrey manuscript that may have been in Welsh or Cornish, but even that bit of evidence cannot be taken on face value, and it returns us to the problem of a now-vanished document that may or may not reflect folk tradition.
There is anecdotal nineteenth-century evidence from writers in Wales and Cornwall who describe examples of people expressing doubt that Arthur would ever return and of being threatened with violence by villagers, and that may hint at how widespread believe in all things Arthur could have been, but it is a giant leap from a few early modern examples to a medieval writer of uncertain origin and of unknown fidelity to tradition.
So given that entirely unsatisfactory overview of the problems, what are we to do with a twelfth century author like Geoffrey of Monmouth? I suspect most people who would have read the author near the time of his writing would have taken the prophecies at face value: that is, they would not regard them as literary invention since that would have employed that level of literary criticism at the time. Frankly, my gut tells me that the prophecies were not literary invention, but rather that Geoffrey was drawing from some sort of tradition. I can't prove that, and certainly someone can write to explain why we should see this as literary invention, but that conclusion would be as speculative as the conclusion rendered by my gut.
If we can accept that most medieval readers would take Geoffrey at his word, that the prophecies were legitimate, then the question is whether they would accept them as accurate. Again, skeptics exist in all societies, so it is given that some, at least, would not believe they were accurate. I suspect that many, however, would accept them as possibly accurate and at the very least intriguing. After all, there are still many people who look to Nostradamus (1503-1566) as a credible prophet of the future, and more recently Edgar Cacye (1877-1945) developed a large following for his prophecies. Because people seem eager to learn from those who profess knowledge of the future, those who arrive with these sorts of insights will be believed by some if not many.
Source: O.J. Padel, ‘Geoffrey of Monmouth and Cornwall,’ Cambridge Medieval Celtic Studies, 8 (Winter 1984), pp. 1–28.
I can't answer whether the contemporary people really believed the Prophecy of Merlin, but can point out here that the tradition, whether in written and in oral form, was far-more wide-spread geographically and had a long afterlife than widely assumed.
The Prophecy of Merlin had been widely circulated in Northern Europe even before Geoffrey’s death (ca. 1155): Abbot Suger of St. Denis, a close advisor of the French royal count, certainly knew its existence in ca. 1140 (Lawrence-Mathers 2012: 177, 234, note 5), and anonymous author of A History of Norway (Historia Norwegie), probably written in Eastern Norway in the third quarter of the 12th century, also refers to King Henry I of England with the epithet from the Prophecy, as quoted as following : ‘He [William the Bastard/ Conqueror] was a father of William Rufus and his brother Henry, who in the prophecies of Merlin is styled the Lion of Justice (Leo justitie)’ [Kunin (trans.) 2001: 9]. It even seems to me from this passage that the author of HN was rather more familiar with the Prophecy itself than William the Conqueror. Not only the Norwegians, but also the Icelanders also show interested in the prophecy of Merlin. Monk Gunnlaugr Leifsson of Þingeyjar in Northern Iceland translated the prophecy, originally in prose form, into Old Norse verses, called Merlínússpá around 1200, based both on the Propechy and Historiae regum Brittaniae in his hand. This Old Norse poem version of the Prophecy, Merlínússpá, was certainly known to some extent in Late Medieval Iceland, has been recently rather famous for the fact that its expressions and tones closely resemble arguably a masterwork for the Norse mythology, Eddic poem Vöruspá (‘Prophecy of the Seeress’) (Kalinke 2011: 5f.). Thus, the (reception study of) the Prophecy was also important to consider the development history of the Old Norse- Icelandic literatures in the 12th and 13th century, though I don’t dare to delve further here in which one was redacted in the extant final form in the parchment after another.
On the other hand, the prophecy of Merlin left its trace mainly in the field of political culture, or political propaganda in High and Later Medieval British Isles as well as other part of Europe. Plassmann points out that the Prophecy was indeed a pioneer work of this genre of political prophecy, i.e. the work with certain political purpose, assuming the name of prophecy in Medieval West. The fame of the Prophecy as well as Merlin the prophet did not diminish immediately with the demise of Angevin Empire, alleged patron of the Arthurian literatures to enhance their claim of legitimacy: To give an example, Merlin succeeded the place of Joachim of Fiore (famous Cistercian mystic-historian: d. 1202) and stood for the Ghibellines (pro-imperial political faction in the Italian city states, like Dante) in late 13th century Italy, with Verus Merlini, but the contents of the work was now not identical to Geoffrey’s original, as Mendicant Salimbene of Palma states (Daniel 2006: 280-282).
Lawrence-Mathers cites even more telling example from Christine de Pisan’s Ditié de Jehanne d’Arc (1429) (Lawrence-Mathers 2012: 190):
‘More than 500 years ago Merlin, the Sibyl and Bede foresaw Joan [Joan of Arc] in their minds and put her into their writings and made prophecies about her as the remedy for France. They said that she would carry the banner in French wars, and exactly predicted her deed’.
So, Merlin was quoted here once again in the 15th century France to legitimate the deeds of Joan of Arc and the French army, against the Englishmen had allegedly served long ago (don’t need to point out the difference between Britannia and England). Despite of this assertion, I can’t find any corresponding passage in Geoffrey’s original(s), though. It is clear from these accounts that the concept of the Prophecy of Merlin was still very popular in Late Medieval West in general, though the fame of the prophecy seemed rather to go ahead of the actual contents so that few people paid attention to the exact content of the original prophesy anymore in course of time.
While it does not necessarily mean that the tradition concerning the prophecy was regarded as true, these popularities of the tradition suggest that the people continued to recognize some sort of the utility either as a political propaganda to legitimate something or as a social criticism, assuming the name of famous prophet, Merlin, throughout the Middle Ages.
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