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.
The words you list all contain what is called an "intervocalic /r/". As danorton mentioned in his answer, in Received Pronunciation an "o" preceding an intervocalic "r" is pronounced as /ɒ/ (like the "o" in "lot" or "orange"). This pronunciation also occurs in Boston, USA. In Canada, the "o" is pronounced /ɔ/ (as in "cord"). In much of the mid-Atlantic (e.g., New York, Philadelphia, and the Carolinas), the "o" is pronounced /ɑ/ (as in "card"). In the remainder of the US, the pronunciation varies between /ɔ/ and /ɑ/ depending on the word. The words you gave as examples are usually pronounced with /ɑ/, whereas words like "horrible", "origin", and "Florida" are usually pronounced with /ɔ/.
In conclusion, this phenomenon varies by dialect. It is also related to the "horse–hoarse merger," in which the vowels /ɔ/ and /o/ are merged when preceding an /r/, thus making words like horse/hoarse, for/four, war/wore, or/oar, morning/mourning, &c., homophones.
Best Answer
The spelling change from 'y' in Middle English to 'i' in Modern English in such words as wife or time is actually a consequence of the phenomenon known as the Great Vowel Shift.
In wikipedia's chart you can follow the path for the sound now in time in the leftmost column. And the corresponding IPA steps are summarised as follows:
Old English
Going back to Old English, the most common spelling for wife and time would be:
For wife wīf and it would be pronounced something like "weef" /wiːf/1 (actually the bar over the 'i' is a modern typographical convention to distinguish long from short vowels as OE does not have this distinction in spelling).
For time tīma and it would be pronounced "teema" /ˈtiːma/1.
The letter 'y' in Old English does exist but it represents the sound /y/ (as in German 'ü' or French 'u'). See for instance lȳtel /ˈlyːtel/ => "little". Wīf insteadis is pronounced with a long i /i:/.
In Old English the spelling wyf would have been a spelling mistake - the correct form being wif.
Middle English
The upheaval triggered by the Norman Invasion, which eventually gave birth to Middle English was marked, among other things, by a change in the spelling conventions. The usage of the letter 'y' was generalised for all words with the sound /i/ or /i:/, thereby following the rules applied in medieval French.
Therefore, the spelling of /wi:f/ as wyf became the rule.
Late Middle English / Early Modern English
Indeed a close examination of the very quote included in the question suggests that the 'y' is pronounced 'ee' as in beauty and not 'eye' as in why. This is visible in some of the words I have highlighted.
As you probably guessed many of the words above are verbatim French spellings. I've checked online for instance the words justyce, felonye and penurye.
Also, keep in mind that the GVS only affected long vowels, so that not all the words above spelled with the letter y are now pronounced with the sound /iː/.
The thing to notice is that, at the time of Thomas Cromwell the normal spelling of wife was wyf (Middle English) or wyfe (Early Modern English) and that it closely matched its pronunciation (/wɪif/).
Modern English
However, the Great Vowel shift was only starting and the next step after passing from OE wīf /wiːf/ to ME wyf /wɪif/ would be to pass to eModE /wəɪf/ and eventually to ModE /waɪf/ wife.
As the pronunciation shifted, so did the the spelling. The most common letter for the diphthong /aɪ/ being the letter 'i' 2 the new spelling for /waɪf/ became our familiar wife.
Note 1 The spelling used for Old English is a new system at the time of king Ælfred. It is based on an extended Latin alphabet and it closely reflects the pronunciation of the time. Although there are inevitably some spelling variants, we are pretty sure of the pronunciation of such common words as wīf and tīma.
Note 2 Looking no further than the first person pronoun 'I'. Interestingly enough this can also be observed in the same quote if one considers the word contenwid (which is actually included twice but with slightly different spellings).