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:
Middle English [iː] diphthongised to
[ɪi], which was most likely followed
by [əɪ] and finally Modern English
[aɪ] (as in mice).
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.
I amongst other have indured a
parlyament which contenwid by the
space of xvii hole wekes wher we
communyd of warre pease Stryffe
contencyon debatte murmure grudge
Riches poverte penurye trowth falshode
Justyce equyte dicayte opprescyon
*Magnanymyte* actyvyte foce attempraunce
Treason murder Felonye consyli …
[conciliation] and also how a commune
welth myght be ediffyed and a[lso]
contenewid within our Realme. Howbeyt
in conclusyon we have d[one] as our
predecessors have been wont to doo
that ys to say, as well we myght and
lefte wher we begann.
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).
It isn't so much a grammatical issue, as an historical one.
To make a very long story short, between the 1300s and 1700s, English underwent a change in pronunciation called the Great Vowel Shift. It was messy, and inconsistent. There are a few theories about why this happened, but there isn't any grand consensus.
Unfortunately for you and every other English language student, the standardization of spelling happened right in the middle, between the 1500s and 1600s. So, many of the spellings were created to reflect older pronunciations, which are no longer used today.
In other cases, the different pronunciations of the letter reflect the language of origin. English is a basically a mishmash of Anglo-Saxon, Norman French, and Latin, with bits of Dutch and Old Norse and others thrown in for fun, based on a long history of conquerors taking over the British Isles.
As it happens, none of the examples you've given in your question are of actual English origin: umbrella is Latin via Italian, and use, utensil, study and student are Latin via French. Other words like up and under are from Old English, while ugly comes from Old Norse. Here, you can already see some patterns, that the words of Latin origin tend to have one sound (use, utensil, student), while the words of Old English / Germanic origin tend to another (up, under, ugly), but even that isn't consistent (study, umbrella). Roughly speaking, these correspond to IPA /u/ and /ʌ/ respectively.
Other examples: great big clusters of vowels tend to point towards a French origin, like oeuvre and bouquet. Clusters of mostly silent consonants tend to point towards very old words of a Germanic origin (brought, knight), where one or more consonants have been elided over centuries of usage.
tl;dr: There are no hard and fast rules for determining the pronunciation of English words directly from spelling. In fact, even native English speakers will occasionally trip over the pronunciation of a particular word, particularly when it is seen written many times before hearing it spoken (from personal experience).
There have been attempts to resolve this issue, which can be fascinating to read about, but as of yet, none have stuck.
In cases of doubt, Wiktionary does a wonderful job of showing pronunciations in IPA format for British and American English, and there is usually at least one audio sample to listen to.
Best Answer
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.