Why did only English undergo the Great Vowel Shift, making pronunciation stray so far from spelling?

I started off by posting a series of comments scattered all over the page, but I thought I should sum them up in a standalone answer.

Generally speaking, there have been similar shifts in many other languages. And they are even happening right now as we speak.

But first things first. Since you mentioned Slavic languages, as the most obvious example, in Old Church Slavonic, an o was an o. In contemporary Russian, it can be anything from a schwa to an ʌ to an ɔ, depending on the position relative to the stressed syllable (e.g. молоко, milk, /məɫɐˈko/ or /məlʌˈkɔ/; водоворот, swirl, /vədəvʌˈro̞t/). Also, in Old Church Slavonic, there were a number of nasal sounds, which are absent in pretty much all contemporary Slavic languages with the notable exception of Polish.

Secondly, don't get me started on German. If you don't know how to correctly pronounce Soest, Troisdorf, Huonker, Pankow, Laermann, Hueck, you will pronounce them wrong. It happens to native German speakers all the time.

Speaking of Germanic languages, the most notable vowel shifts happened in German and Dutch (Wikipedia even mentions them in the article on the Great Vowel Shift). It's just that there was at least some concerted effort to keep the spelling consistent with the (changing) pronunciation. The pronunciation shifts were accompanied by spelling shifts, if you will. Hence the popular but wrong assumption that there weren't pronunciation shifts to begin with.

In other words, what made the pronunciation stray so far from spelling in English was not the Great Vowel Shift; it was the absence of the accompanying Great Spelling Update.

Now, it's always a bit harder to explain the absence of something rather than its presence, though one of the other answers does provide an interesting link. On a more general note, I will say that spelling reforms are the domain of politicians, one of the most prominent and recent examples being the German orthography reform of 1996, kicked off by the Conference of Ministers of Culture and later monitored by the International Commission for German Orthography. English, however, traditionally lacks such regulatory bodies.

Anyhow, vowel shifts happen all the time, especially on the dialect level. Now that I have zeroed in on German, I'll just take Bavarian as an example. In Bavarian, viel is not pronounced as /fiːl/ and ein Haufen is not pronounced /aɪ̯n ˈhaʊ̯fm̩/. But again, there is some effort to keep the spelling consistent with the pronunciation, so if you came up with the crazy idea to write a Wikipedia in Bavarian, you would spell viel as vui, vei, vii or fui, and ein Haufen as a Haufa, to reflect the actual pronunciation. And there are also Wikipedias in Ripuarian, Plattdeutsch, Alemannic... It's hard to imagine, say, standalone Australian, Canadian or Texan Wikipedias where the spelling mirrors the local dialect in such a manner.

I guess I can sum these ramblings up as follows: vowel shifts happen all the time. Spelling conventions are a question of politics and culture.

Only Middle English underwent the Great Vowel Shift because that name specifically refers to a historical shift in English. Other languages have had shifts, either unnamed or of different names. Notable named examples are the first Germanic sound shift, the second Germanic sound shift, and the northern cities vowel shift.

The Great Vowel Shift seems to be more prominent than it really is because of the peculiarities which it introduced into spelling, but this isn't a feature of English so much as it is a historical accident of a shift occurring after the introduction of the printing press. English being one of the most-studied language in historical linguistics magnifies the apparent significant of the Great Vowel Shift too, and the grandiosity of the name can be attributed to that more than anything else.

Well, there was something that might have been called 'vowel shift' in Polish. A long, long time ago probably about 400 years back, We had a long-short vowel distinction in Polish, also one nasal vowel had a different timbre. 'Ą' wasn't a nasal 'o' [õ] like in contemporary Polish, but more like a nasal 'an' [ã] found in French. As for consonants, ć, ś, ź, dź were half-palatalized; more like the Russian [sʲ, zʲ, tsʲ,dʲ]. The letter 'ł' sounded like the English consonant called 'dark L' [ɫ] but it got vocalized to [w]. And last but not least, the 'y' that represents the IPA [''I] was closer to the Australian English Kit vowel [ +I] (very front).