So which pronunciation is standard for the [ʊ] sound? Rounded or unrounded?
Certainly there is some rounding, but because roundedness is not phonemic in this position, there is also considerable variation in how much of it actually occurs in any given word and speaker.
For example, you will find that it is generally somewhat more rounded in pull and full than it is in put and foot respectively. That’s because having an r or an l right next to it rounds it off a bit — which is why it is a bit more rounded in root and rook than it is foot or cook. Same with rookie versus cookie, where the first version is a bit more rounded than the second. And of course, a w helps: compare how wool is even more rounded than full, and also moreso that wood.
I believe English has no words with [ʊw], as that seems redundant. However, it can occur in phrases, especially in some dialects, where something like I knew it full-well may approach that.
However, it is still perceived as the very same phoneme in all those words and cases I’ve just listed above.
Correction — or not
I said that I thought English had no words with [ʊw] in them. And at the end of the day, I still believe that. However, I have discovered that grepping the OED yields the apparent existence-proof counterexample of Rauwiloid, which means:
A proprietary name for a hypotensive preparation containing a number of alkaloids extracted from Rauvolfia serpentina.
You also have compound words whose first element ends in [aʊ] (rather than [aw], as it is sometimes spelled) connecting to something that begins with [w], and which have in effect a “double w” in them, you expand the list to include such things as:
bow-wow, powwow, skeow-ways, wow-wow
Finally, if you consider the sound in words like no and micro to be
an [oʊ] diphthong rather than [ow], then you get all these, most of which were originally compounds of some sort:
froward, frowardly, frowardness, glow-worm, Holloway,
hollowwort, Howeitat, Khowar, meadow-wink, microwave, microweld,
Moldo-Wallachian, nowise, Oldowan, Parowax, powan,
shalloway, slow-worm, swallowwort, werowance,
yellow-wood, yeowoman.
For example, yeowoman theoretically yields /ˈjoʊwʊmən/, at least in North America. Still, there is a reasonably convincing argument to be made that that one is better written as simply /ˈjowʊmən/.
Slightly less uncommon is nowise, which is a compound of one word ending in a diphthong connected to another starting with a triphthong, so /ˈnoʊˌwaɪz/.
But I am still highly dubious of the existence of [ʊw], because I think it fuses into the semi-consonantal glide, [w]. After all, nowise and no eyes are homophonic, so I think this idea of [ʊw] is very hard to justify, and so I stand by my initial statement.
Even towel is usually pronounced with just one syllable, /taʊl/, thereby rhyming with cowl /kaʊl/. Even with folks who work very hard to put two syllables into that, with /ˈtaʊ.wəl/, I submit that you could write that /ˈtawːəl/ and avoid the whole controversy of whether a semi-vowel/semi-consonant/off-glide is really /ʊ/ or really /w/. However you write it, it seems like the same sound to me, such that bisyllabic towel just has a geminate [w]: /ˈtaw.wəl/.
Best Answer
My intuition is that these three pronunciations are in free variation, so there is no morphophonemic rule that can help you decide when to use which. The first, where /n/ and /t/ are both pronounced you can call a "recitation" form, and you would use it, for example, when dictating a message over telephone or speaking to a foreigner who has trouble hearing English.
The other two are gotten by two common rules of reduction. First is assimilation of the nasal and plosive consonant (t not pronounced). Second is loss of the nasal consonant and nasalization of the vowel. Unstressed /t/ preceding a vowel may be flapped, so that's why you get the flap pronunciation.