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
Particular instances of the expression "everything's coming up X" for the past 60 years are heavily influenced by the hit song "Everything's Coming Up Roses," from the 1959 Broadway musical (and subsequent 1962 film) Gypsy, about the stripper Louise Hovick (aka Gypsy Rose Lee), whose mother's name was Rose.
The simple sense of the expression is "everything is great" or "everything is going to come out well." Nigel Rees, A Word in Your Shell-like (2004) offers this brief discussion of the term:
Early published instances of 'everything's coming up roses'
A Google Books search for "everything's coming up" for the period 1900–1965 yields no matches before 1959—and the first matches for that year are from "Reviews of This Week's LP's*," in Billboard (November 2, 1959):
Not only does Billboard praise two versions of the song from Gypsy, but it cites what may be the earliest verbal play on the song title—Van Damme's LP title, Everything's Coming Up Music.
Early instances of 'everything's coming up [something besides roses]'
A year later, the replacements for "roses" in the expression become more frequent. From Broadcasting, volume 59 (1960) [combined snippets]:
From Historic Garden Week in Virginia (1960) [text not viewable in snippet window]:
From an article in Harper's Bazaar, volume 96, part 1 (1963) [combined snippets]:
From an article in The Iron Age, volume 196, issues 19–26 (1965) [combined snippets]:
And from an unidentified article in Town & Country, volume 119, issue 4521 (1966) [combined snippets]:
Early instances of 'everything's coming up roses' as an idiom
Meanwhile, back in 1960, we find the expression "everything's coming up roses" being applied as an idiomatic way of saying "everything is going great" to the plastics industry, in an unidentified article in The Rubber Age, volume 88 (1960[?]) [combined snippets]:
From an article in Forbes magazine, volume 96 (1965) [combined snippets]:
And from an advertisement for Automation magazine in Industrial Marketing, volume 51, issues 7–12 (1966) [text not shown in snippet window]:
Antecedents to 'everything's coming up roses'
Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) offers this entry for the expression "come up roses":
In Google Books and Elephind searches, the oldest expression seems to be "come out smelling like a rose [or roses]." From an untitled item in the Salt Lake [City, Utah] Herald (July 19, 1885):
From S.J. Osten, "From District Council No. 9" (February 26, 1912), in The Blacksmiths Journal (March 1912):
From Clyde Davis, Sullivan (1940) [combined snippets]:
From an unidentified short story in Story, volume 32 (1948):
From "Fearful of Votes Back Home," in the Indianapolis [Indiana] Times (March 29, 1949):
And from "Like a Rose?" in Naval Aviation News (March 1953):
If nothing else, these early instances of "come out [or up] smelling like roses [or a rose]" tend to confirm Partridge's view that the noteworthy aspect of smelling like a rose in the original expression was that it was preceded by falling into something extremely noisome.
Conclusions
Expressions of the form "everything's coming up X" almost certainly owe their existence to the song "Everything's Coming Up Roses" from the musical Gypsy (1959). Whether it, in turn, was influenced by the earlier expression "come out [or up] smelling like roses [or a rose]" is less clear, but it doesn't seem far-fetched to me. The latter expression has been in use since at least 1885.