To broaden the discussion of pronunciation possibilities a bit, I ran a Google Books Ngram search for "an ouroboros" (the blue line), "a ouroboros" (no line), "an uroboros" the green line), and "a uroboros" (the red line), with the following results:
The red line (which reflects some two dozen matches for "a uroboros" between 1951 and 2008) suggests that at least some English speakers pronounce the first syllable of uroboros as yo- or yoo- or yoor-, or yor-. A separate Google Books search for "a ouroboros" turns up seven matches for that spelling, as well (although the number was evidently too small to support a plotted line in the Ngram graph). For example, from The British Museum Quarterly, volume 31 (1966):
The Cairo example stands within an oval formed by a ouroboros snake, a motif which, rare in the art of dynastic Egypt, becomes a common design on the magical amulets from Roman Egypt.
And from Pierre François, Inlets of the Soul: Contemporary Fiction in English and the Myth of the Fall (1999):
The alchemy-related metaphor in the excerpt above affords one of the most far-reaching glimpses of the "premises" normally hidden from human senses. The "serpent-craft around Jupiter, Venus" is a hierogamy—a divine wedding of the cthonic (serpent) and heavenly (Jupiter, Venus) deities—a ouroboros image identifying the circular periphery and the centre of the cosmic fabric.
And from Michael Speyer, The Kitish (2009):
Judas had had no wife or children, and had cut all ties with his previous life so that his family would not suffer Roman retribution for his ations; he had exchanged family life for a cause in which he had believed without reservation, that i, until he had encountered the Master, who had made him see that strife and conflict were of the nature of this world, and that to blindly follow any cause no matter how noble was to throw away one's life for something that had no resolution, and which would keep repeating itself throughout the ages like a Ouroboros.
My assumption here is that use of an as the indefinite article before ouroboros or uroboros implies pronunciation of the noun with a y sound at the beginning of the word, whereas use of a as the indefinite article implies pronunciation of the word without a y sound at the beginning of the word.
The Ngram chart strongly suggests that pronunciations that include a y sound at the beginning of the noun ouroboros or uroboros are less common than pronunciations that begin with an o-, oo-, ou-, or-, oor-, and our- sound (treated collectively). Nevertheless, the chart results indicate that pronunciations of the noun that begin with a y sound do (probably) exist among English speakers.