Spelling rules for V and U
In terms of spelling conventions, the modern convention in texts that distinguish V and U is to use V only for a non-syllabic sound that starts the syllable it's in. So V is used in vos and servus but not in ultra (where U is syllabic), or in lingua, aequa, suadeo, nauta (where U is not the first sound in the syllable). This is basically equivalent to using V in contexts where the traditional English, French and Italian pronunciations of Latin generally used the sound [v]. Other spelling conventions have previously been used; e.g. one common convention in older printed material is to use v at the start of words, u elsewhere, regardless of whether the sound is syllabic or not (so vos, seruus, vltra, lingua, nauta).
Classical Latin had no spelling distinction between V and U: the same sign was used for both (in inscriptions, it looked basically like V), aside from some rare uses of Ⅎ, a new letter introduced by Emperor Claudius for [w] that didn't catch on.
Latin phonology of V/U
The question of what we're dealing with in terms of Classical Latin pronunciation and phonology is not too clear. In Latin, as in English, there is no possible phonemic contrast between the diphthong /au̯/ and the vowel-glide sequence /aw/ before a consonant. There is a contrast with /a.u/, a sequence of vowels in hiatus (pronounced in two separate syllables), although the hiatus sequence /a.u/ is rare.
It is common to analyze /w/ as not occurring before a consonant. In that case, you can conclude that the disyllable auspex starts with the diphthong /au̯/.
However, in theory, you can analyze "auspex" as starting with /aw/, and in fact, this analysis has been defended. The putative diphthong "au" is analyzed as /aw/ in "Aspects of the Phonology and Morphology of Classical Latin" (2016) by András Cser ("2.2.2. The question of diphthongs", pages 31-37).
AV before a vowel shows a problematic contrast in behavior between words like CAVVS and words like AGAVE. The former has a light syllable, indicating the syllabification /ka.wus/. The latter, from Greek Ἀγαύη, shows a heavy second syllable:
'ādspĭcĕ|, mātĕr!' ă|īt. vī|sīs ŭlŭ|lāvĭt Ă|gāue
(Ovid Metamorphoses 3.725, dactylic hexameter, macrons and breves marking syllable weight not vowel length)
Some dictionaries mark the vowel in AGAVE with a macron, which could suggest it is pronounced with /aːw/ as in suāvis /swaː.wis/. However, it is a little difficult to understand why the Greek diphthong αυ would be adapted to a Latin long /aː/ sound followed by /w/. Therefore, some sources on Latin pronunciation indicate that the AV in words like AGAVE was in fact pronounced differently in some way from the AV in words like SVAVIS.
Some possible ways to analyze the contrast between CAVVS and AGAVE are as follows:
/ka.wus/ vs. /a.gau̯.eː/: This analysis implies that the diphthong /au̯/ in in fact distinct from the sequence /aw/, although they do not contrast before consonants. This analysis is given in "Antevocalic u-Diphthongs in Latin", by Henry M. Hoenigswald (Language
Vol. 25, No. 4 (Oct. - Dec., 1949), pp. 392-394, accessed though JSTOR).
/ka.wus/ vs. /a.gaw.eː/ This analysis would violate the otherwise very consistent rule in Latin that a consonant is always syllabified with an immediately following vowel. I'm not sure if I've seen it defended anywhere.
/ka.wus/ vs. /a.gaw.weː/: This analysis is consistent with Latin phonotactic rules, although /w.w/ does not occur in Latin outside of words from Greek with αυ or ευ before a vowel. It's similar to the usual analysis of words like maior as containing geminate /j.j/. It requires explaining why Greek αυ followed by a vowel would be perceived by Latin speakers as /aw.w/.