Assuming we take the symbol "ʌ" to represent more or less the STRUT vowel, some contemporary North American speakers have a diphthong more or less pronounced [ʌɪ] (it could also be transcribed as [əɪ], [ɜɪ], or [ɐɪ]) as an allophone of /aɪ/. For certain other North American speakers, there may be a marginal phonemic split between /ʌɪ/ and /aɪ/.
This is because of the phenomenon called "Canadian raising". Basically, [ʌɪ] is regularly used instead of [aɪ] before voiceless/fortis coda consonants. But for many speakers, [ʌɪ] can also show up in some other contexts.
In North American English, /t/ and /d/ are often lenited in certain "weak" positions, which causes them to become the voiced flap [ɾ] before a vowel. For some speakers with t-lenition, there is a phonemic contrast between [ʌɪɾ] (with [ɾ] from lenited /t/) and [aɪɾ] (with [ɾ] from lenited [d]). The classic example of this contrast is "writer", with [ʌɪ] like the verb "write" [ɹʌɪt], vs. "rider", with [aɪ] like the verb "ride" [ɹaɪd].
If we assume that /t/ and /d/ actually remain distinct phonemes intervocalically, despite the apparent or near-neutralization to [ɾ], then the phonemic distinction between "writer" and "rider" can be explained as a difference in the medial consonant phoneme rather than as a difference in the vowel phoneme.
But there are some speakers who use [ʌɪɾ] in certain monomorphemic words that historically contained the phoneme /d/, such as "spider". And I've heard that some speakers even have [ʌɪ] before voiced consonants other than [ɾ] in certain monomorphemic words, such as tiger. I think speakers who use [ʌɪ] in words like spider or tiger may be more likely to think of /ʌɪ/ as a distinct phoneme from /aɪ/.
A diphthong something like [ʌɪ] is also commonly supposed to have been a step in the development of [aɪ] from original [iː]; e.g. something like [iː] > [ɪi] > [əi] > [ʌi] > [ai]. However, this never contrasted with any preexisting /ai/ phoneme, because Old English /ai/ was already merged with /ei/ to something like [ɛi~æi] (the "vein-vain merger"). (I don't know how strong the evidence is for the exact steps given above, versus e.g. "[iː] > [ɪi] > [ei] > [ɛi] > [ai]" with the reflex of OE /ai/ and /ei/ having already become monphthongal /eː/ by the time the reflex of /iː/ was lowered to something like [ɛi].)
I don't know of any dialects, past or present, that have two distinct diphthongs /ʊɪ/ and /uɪ/, so I'll deal with them together. A diphthong [ʊɪ~uɪ] is generally supposed to have existed in Middle English, corresponding to some cases of Modern English /ɔɪ/: see English Language and Linguistics Online "Diphthongs", which says
The Middle English [oi] was [ɔi] in words like noise and royal, and remained until today. However, there are also some words pronounced [oi] that derived from ME [ui]. This diphthong developed into [əi] and changed to [ai] in the Early Modern period. It occurred in words like boil, destroy, join etc. In PDE it is pronounced [ɔi]. (Cf. Barber 1976: 304)
My impression is that it was mainly spelled the same as /ɔɪ/, so "oi/oy" etc.
Roger Lass says
The diphthongs /oi/ (joy, choice) and /ui/ (join, poison), though usually spelled alike in Middle English, were nevertheless kept apart—if not always according to etymology—until well into the eighteenth century. Hart regularly writes <oi> for /oi/ and <ui> for /ui/, and has an occasional third value written <uei> = /wɛi/ in a few words like the Dutch loan buoy. Hodges (1644) still retains two sets: one apparently has [ɒi] and the other [wɛi] (boy, choice, joy vs. boil, coin, point). Wallis has [ɒi] in boys, noise, toys and -- probably -- [əi] in boil, oil, toil; but he notes that the latter set can also have [ɒi]. And [əi] is Wallis' usual reflex for ME /iː/, so there is a partial merger which we can exemplify by loin and line.
(A History of the English Language, "Phonology and Morphology," p. 88, edited by Richard Hogg and David Denison)
I also found discussion of this in Written Language: General Problems and Problems of English, by Josef Vachek.
Vachek (p. 41-44) says that the first, nuclear vowel in /ui/ developed “perfectly parallel” to the vowel in much and fun, and transcribes the stages of its development as “oi”and “əi”, saying it would have reached the latter by the 18th century. Vachek apparently uses <ə> to represent the quality of a stressed “short u” sound (he later identifies ə as the vowel of come and love), so I would re-write this as /ui/ > /ʊi/ > /ʌi/; the notation doesn’t really matter though.
The evidence for this development seems to be the merger with the reflex of Middle English /iː/ mentioned by Lass: Vachek provides the example of rhymes between “joins—refines, toil—compiles, and poison—surprise on and the like” (p. 41) from poems in the 17th and 18th centuries and explains these as having əi as the nuclear vowel: in the first member of the pair from /ui/, and in the second member of the pair from /iː/. The development that Vachek gives for /iː/ is ɪi > ei > əi > ai, with the stage əi reached by the 17th century.
He says the merger of /ui/ words with CHOICE rather than PRICE in present-day English is “commonly explained as due to the influence of the spelling” (p. 42), and he doesn’t think this is completely off-base, but he expresses dissatisfaction with this explanation in light of the fact that words such as come and love retain the irregular spelling-pronunciation correspondence “o” = ə. He argues that the pushback against merging /ui/ with PRICE was “functionally motivated” by a tendency to mark words of non-native origin with markedly non-native sounds, like the CHOICE diphthong, rather than unmarked native sounds like the PRICE diphthong (p. 43).
Vachek’s argument is not extremely convincing to me. Come and love are very frequent words, so I think they would be expected to resist spelling pronunciation more than most other words. It seems to me that there probably are some less frequent words spelled with “o” that were originally pronounced with the STRUT vowel and then developed spelling pronunciations with LOT or GOAT: for example, certain words such as wort and wont; certain words with <o> from French <o~u~ou> such as pommel, conduit, conjure, colander (compare compass, comfort, company, money, stomach, color); and miscellaneous words of uncertain origin and variable pronunciation (which admittedly might not be examples of spelling-pronunciation) such as donkey (possibly derived from “dun” or “Duncan”).
I remember reading somewhere that some people don't think there was ever really a phonemically distinctive contrast between [ʊɪ~uɪ] and [oɪ~ɔɪ~ɒɪ], even in Middle English (note that the distinction that Lass talks about seems to have been realized in various ways, often involving a merger of one or the other sound with a third sound or sequence of sounds), but I don't really understand what the argument for this position is.
Geoff Nunberg made a comment beneath the Language Log post "Who would not weep, if E. B. White were he?" that seems to suggest that the words with a vowel that could be merged into the "long i" sound tended to have certain types of surrounding consonants:
consider the 18th c. blurring of the nuclei of words like line and loin, which turns up several times in the Essay of Criticism:
One notable point here is that this confusion was phonetically conditioned, limited to vowels before /n/, /l/ and some other sonorants, particularly when preceded by /p/ and /b/ (as in point and boil). Nobody rhymed toy and sigh as far as I know. So the rhymes here turned on a perception of phonetic closeness or identity, not simply a rhyming convention. A second is that the confusion left several doublets in its wake (rile and roil, for example) as well as some dialect variation: Dickens had his lower-class characters saying spile for spoil and jint for joint. A third is that the confusion was noted, and sometimes criticized as "abusive," by contemporary writers.
I found the partial overlap between the words Nunberg mentions and the list of words from Barber as reported by English Language and Linguistics Online to be interesting: Barber mentions the word destroy, which doesn't have a sonorant consonant after the diphthong, but the words boil and join occur in both lists.
A somewhat comparable situation may exist in some recorded forms of Brooklyn English. Accents of English: Volume 3, by John Wells, reports on work by William Labov that found that certain Brooklyn English speakers used [ɜɪ] for /ɔɪ/ in certain words such as voice and join (but "only in the environment of a following consonant belonging to the same morpheme"). Wells says he is not in favor of analyzing this as a separate phoneme. I don't know if the distinction has any direct relation to the old /ui/ and /oi/ sets, but I thought that it seems somewhat relevant even if not.
The word "buy" isn't, as far as I can tell, part of the standard /ui/ set that Lass is talking about. The words that Lass categorizes as having /oi/ and /ui/ are all loanwords (mainly from French/Romance languages); buy is a native Germanic word. It looks like Hart may have transcribed this word with "ei", his symbol for the reflex of Middle English [iː]: John Hart's pronunciation of English (1569-1570), by Otto Jespersen, suggests that "beiër (= buyer?)" (apparently there is some uncertainty). I give my best guess as to why this word is spelled with "uy" in my answer to Why is "build" spelt with a "u"?
Basically, Old English had a high front rounded vowel [yː] (mostly written "y") that developed in most accents, including the one that is the main ancestor of Modern English, to an high front unrounded vowel [iː] (mostly written "i" or "y"), but to something different in some other accents that was written "uy" or "ui". I have never seen it written that this "something different" was a diphthong like [uɪ], though: it may have been a monophthong that just happened to be written with a digraph. I don't know if we have any detailed information about the sound of this "uy/ui"; although the vowel [uː] in the word "bruise" in present-day English seems to have developed from something similar.
cuisine, biscuit, circuit, conduit
I am not too familiar with the evolution of this set of words. The "ui" in Old French conduit originates from vocalization of the consonant /k/ in Latin conductus, which seems to suggest that it was originally pronounced with a falling diphthong, but at some point in French all falling diphthongs were converted to rising diphthongs.
In English, I don't think these words ever fell into the /ui/ set that Lass mentions, although I'm not entirely sure. The pronunciations in present-day English all represent "u" = /w/ (the loss of "w" after a consonant in an unstressed syllable was a semi-regular sound change; compare answer, conquer, Greenwich), aside from conduit, which has picked up what seems to be a spelling-pronunciation with /dju.ɪt/ (or some variant of that like /djʊ.ɪt/, /du.ɪt/, /dʒu.ɪt/).